\documentclass[11pt,reqno]{amsart}
%\usepackage{inputenc}
\usepackage[english,russian]{babel}
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{amsfonts}
\usepackage{graphicx}
\usepackage{xcolor}
\usepackage[colorlinks]{hyperref}



\usepackage{titlesec}
\titlespacing{\section}{0pt}{2.5ex}{1.5ex}
\titlespacing{\subsection}{0pt}{1.5ex}{1ex}
\titlespacing{\subsubsection}{0pt}{1.5ex}{1ex}
\titleformat{\section}{\large\bfseries\centering}{\thesection}{1em}{}
\titleformat{\subsection}[runin]{\bfseries}{\thesubsection.}{0.5em}{}[.\mbox{\ }]
\titleformat{\subsubsection}[runin]{\bfseries}{\thesubsubsection.}{0.4em}{}[.\mbox{\ }]

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\usepackage{tikz}
\usepackage{float}
\usepackage{url}
\usepackage{mathtools}
\usepackage{amsmath,amsfonts,amsthm}
\usepackage{mathrsfs} 
\newcommand{\irr}{\operatorname{irr}}
\newcommand{\dist}{\operatorname{dist}}
\newcommand{\case}[1]{\paragraph*{Case #1:}}
\newtheorem{hypothesize}{Hypothesize}
%=================================
\newcommand{\Gut}{\operatorname{Gut}}


\newtheorem{lemma}{Lemma}
\newtheorem{theorem}{Theorem}
\newtheorem{definition}{Definition}
\newtheorem{corollary}{Corollary}
\newtheorem{proposition}{Proposition}

\newtheorem{example}{Example}
\newtheorem{remark}{Remark}
\newtheorem{question}{Question}
\newtheorem{observation}{Observation}
\newtheorem{conjecture}{Conjecture}


\usepackage{subfig}

\DeclarePairedDelimiter\ceil{\lceil}{\rceil}
\DeclarePairedDelimiter\floor{\lfloor}{\rfloor}

%==================Packages=========
\usepackage{amssymb}

%===================Commands ======
\DeclareMathOperator{\DS}{DS}
\newcommand{\SG}{\mathrm{HGut}_s}
%\DeclareMathOperator{\SG}{SGut(G)}
\newcommand{\HGut}{\mathrm{HGut}}
\newcommand{\SN}{\mathcal{S}_n}
\newcommand{\PN}{\mathcal{P}_n}
\newcommand{\TGut}{the Exponential Gutman index}
%=========================

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%




%%%%%---------- ORCiD
\newcommand\orcidicon[1]{\href{https://orcid.org/#1}{\includegraphics[scale=0.02]{orcid.pdf}}}


\def\udcs{519.172.1} %Here the author places classificators of the paper according to Russian classification system
\def\mscs{05C09, 05C35, 05C07, 05C12, 92E10, 68R10.} %Here the author places  classificators according to the AMS classification list.
\setcounter{page}{144}




\hypersetup{linkcolor=blue,filecolor=black,urlcolor=blue, citecolor=blue}

\def\logo{\textcolor{blue}{{\bf\Huge S\raisebox{0.2ex}{\hspace{0.55ex}\raisebox{0.05ex}e\hspace{-1.65ex}$\bigcirc$}MR}}}

\def\semrtop
     {
  \vbox{
     \noindent\logo \hspace{-3mm}  \parbox{12cm}{     \begin{center}
     {\LARGE \textsf{СИБИРСКИЕ ЭЛЕКТРОННЫЕ}} \\[2mm]
     {\LARGE \textsf{МАТЕМАТИЧЕСКИЕ ИЗВЕСТИЯ}} \\[2mm]
     {\Large \textsf{Siberian Electronic Mathematical Reports}} \\[1mm]
     {\Large\tt{\textcolor{blue}{http://semr.math.nsc.ru}}}\\[1mm]
     \raisebox{1ex}{ISSN 1813-3304 }
     \vspace*{-3mm}
%     {\small Èíñòèòóò ìàòåìàòèêè èì. Ñ.Ë.Ñîáîëåâà ÑÎ ÐÀÍ}\\[-1mm]
%     {\small Sobolev Institute of Mathematics SB RAS}
     \end{center}
}
     \vspace{-1mm}
     \noindent
      \begin{tabular}{c}
     \hphantom{aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa!} \\
     \hline\hline
     \end{tabular}

     \vspace{3mm}
     {\flushleft\it Vol. 20, No. 2, pp. 144--145 (2023) \hspace{\fill}{\rm\small УДК \udcs \hspace{-4mm}}}
     \newline  %Volume, pages in Russian, filled by Editorial board
     	{\rm\small \textcolor{blue}{https://doi.org/10.33048/semi.2023.20.???}}\hphantom{aaaaaaaaaaaaaaaaaaaaaaaa\;}{\rm\small MSC\ \ \mscs }
  }%\vbox
}




\begin{document}
\renewcommand{\refname}{References}
\renewcommand{\proofname}{Proof.}
\renewcommand{\figurename}{Fig.}




\thispagestyle{empty}

\title[On the Inverse Problem]{\large On the Inverse Problem of the Exponential Gutman Index of Trees}


\author[J. HAMOUD]{{\bf J. Hamoud}\protect\orcidicon{0009-0002-0192-3627}}



\address{Jasem Hamoud  % Êîíòàêòíûå äàííûå âñåõ àâòîðîâ è ìåñòî ðàáîòû óêàçûâàþòñÿ òîëüêî íà àíãëèéñêîì ÿçûêå
\newline\hphantom{iii} Moscow Institute of Physics and Technology,
\newline\hphantom{iii} 9 Institutskii Lane, 
\newline\hphantom{iii} 141700 Dolgoprudnyi, Russia}%
\email{\textcolor{blue}{hamoud.math@gmail.com}}%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\author[D. ABDULLAH]{{\bf D. Abdullah}\protect\orcidicon{0009-0008-6855-1729}}



\address{Duaa Abdullah  % Êîíòàêòíûå äàííûå âñåõ àâòîðîâ è ìåñòî ðàáîòû óêàçûâàþòñÿ òîëüêî íà àíãëèéñêîì ÿçûêå
\newline\hphantom{iii} Moscow Institute of Physics and Technology,
\newline\hphantom{iii} 9 Institutskii Lane, 
\newline\hphantom{iii} 141700 Dolgoprudnyi, Russia}%
\email{\textcolor{blue}{abdulla.d@phystech.edu}}%


\thanks{\sc Hamoud, J., Abdullah, D.,
On the Inverse Problem of the Exponential Gutman Index of Trees}
\thanks{\copyright \ 2026 Hamoud, J.}
%\thanks{\rm The work is supported by RFFI (grant 03-01-11111)}
\thanks{\it Received January, 1, 2023, Published December, 31, 2023}%


\semrtop \vspace{1cm}
\maketitle {\small

\vspace{-12pt}
\centerline{{\it Communicated by} {\sc P.P. Petrov}
%({\it filled by Editorial board})
}

\bigskip
\bigskip

\begin{quote}
\noindent{\bf Abstract:} This paper investigates the extremal behaviour of the exponential Gutman index $$\operatorname{HGut}(G) = \sum_{uv \in E(G)} \bigl(2^{d(u)d(v)} + dist(u,v)\bigr),$$
over the class of trees on $n$ vertices. The path graph $\mathcal{P}_n$ uniquely minimises $\operatorname{HGut}(G)$
among all trees on $n \geqslant 5$ vertices. We identify the balanced double star $DS\!\bigl(\lfloor n/2\rfloor,
\lceil n/2\rceil\bigr)$ as the unique maximiser, by establishing a sharp upper bound on the degree product of any edge in a tree.
A closed form expression is derived for complete binary trees, characterizing extremal graphs, addressing the inverse problem graphs realizing a given value $\kappa$.
We further establish a lower bound through Jensen's inequality and an upper bound through the Moore bound. 

\medskip

\noindent{\bf Keywords:} Exponential Gutman index, 
topological indices, 
extremal graph theory, 
degree product, 
threshold graphs, 
binary trees, 
Zagreb indices, 
chemical graph theory.
 \end{quote}
}

\bigskip

\section{Introduction}\label{sec:intro}

Graph theory provides powerful tools for modeling molecular structures, and topological indices have been widely used for decades to the concept of  predict 
physicochemical properties of chemical compounds. 
The foundational work in this area includes the Wiener index \cite{Wiener1947}, which correlates well with boiling points of alkanes through the sum of distances between all pairs of vertices in the molecular graph.

Let $G=(V,E)$ be a simple connected graph with $n=|V(G)|$ and $m=|E(G)|$. For vertices $u,v\in V(G)$, let $d(u,v)$ denote the distance (length of a shortest path) between $u$ and $v$, and let $d(v)$ denote the degree of vertex $v$. The Wiener index is defined as
\[
W(G) = \sum_{\{u,v\}\subseteq V(G)} d(u,v).
\]
A natural degree-weighted extension of the Wiener index is the \emph{Gutman index}, introduced by Gutman in 1994 \cite{GutIndex} as a refinement of the Schultz index \cite{Schultz1989}. It is given by
\begin{equation}\label{eq:gutman}
\Gut(G) = \sum_{\{u,v\}\subseteq V(G)} d_G(u)\,d_G(v)\,\mathrm{dist}(u,v).
\end{equation}
Furthermore; see papers~\cite{reef4,GutIndex,Mazorodze2014}
In recent years, various exponential versions of classical topological indices (such as the exponential Wiener index) have attracted considerable attention. Following this direction, we introduce the \emph{Exponential Gutman index} $\HGut(G)$, defined as
\begin{equation}~\label{eqq01hgut}
\HGut(G) = \sum_{uv\in E(G)} \bigl(2^{d(u)d(v)} + \dist(u,v)\bigr).
\end{equation}

Note that for every edge $uv$, we have $\dist(u,v)=1$, so the definition can equivalently be written as
\[
\HGut(G) = \sum_{uv\in E(G)} 2^{d(u)d(v)} + m.
\]
In this paper, we consistently use definition \eqref{eqq01hgut}.
Several generalized variants of the Gutman index have also been studied. One common generalization takes the form
\[
\Gut_{\alpha,\beta}(T) = \sum_{\{u,v\}\subseteq V(T)} [d_T(u) d_T(v)]^\alpha [\dist_T(u,v)]^\beta, \quad \alpha,\beta \in \mathbb{R},
\]
for trees $T$. Extremal results for such indices on trees are known under various constraints on $\alpha$ and $\beta$ \cite{Das2015, Andova2012, Mukwembi2012, reef10}.

The Zagreb indices, initially introduced in \cite{Gutman1973} and \cite{new3}
in 1972 and 1974, represent some of the earliest and most extensively investigated
degree-based topological indices \cite{Todeschini2010}. For a simple connected graph
$G = (V,E)$, these indices are defined as
\[
M_1(G) = \sum_{v \in V} d_G(v)^2 \hspace{3mm} , \hspace{3mm}
M_2(G) = \sum_{\{u,v\} \in E} d_G(u) \cdot d_G(v).
\]

The main objective of the present paper is to investigate the exponential Gutman index $\HGut(G)$, with a focus on deriving sharp extremal bounds in terms of classical invariants ($n$, $m$, $\Delta$, $d$, Wiener index, Zagreb indices), characterizing extremal graphs, addressing the inverse problem (graphs realizing a given value $\kappa$). We study extremal values of this index among trees of order $n$, examine its behavior on specific families of graphs, and explore some of its basic properties. 

We present our main results in three parts. 
First, we derive sharp bounds connecting the Gutman index with other well-known topological indices. 
Next, we determine extremal graphs with respect to the Gutman index under fixed degree sequences. 
Finally, we address the inverse problem concerning graphs that are uniquely determined by their Gutman index.

%==================================
\section{Preliminaries}\label{sec:prelim}
%==================================
All graphs considered are simple, connected, and undirected.
A \emph{tree} on $n$ vertices has exactly $n-1$ edges.
The \emph{path} $\PN$ and the \emph{star} $K_{1,n-1}$ are the two extremal trees
with respect to many degree-based and distance-based indices.
The \emph{double star} $\DS(a,b)$ (with $a,b\geqslant 1$ and $a+b=n$) consists of two
adjacent vertices (the \emph{centres}) of degrees $a$ and $b$, respectively, with
$a-1$ pendant neighbours of the first centre and $b-1$ pendant neighbours of the
second; it has $n=a+b$ vertices and $n-1$ edges.
Observe that $\DS(n-1,1) = K_{1,n-1}$ (the star) and $\DS(1,n-1)$ is its mirror.
As usual, $d(v)$ denotes the degree of a vertex $v$, and $d(u,v)$ denotes the distance between vertices $u$ and $v$. We denote by $\SN$ the star on $n$ vertices and by $\PN$ the path on $n$ vertices.
For the classical Gutman index, the following extremal results on trees are known: the star $\SN$ minimizes and the path $\PN$ maximizes the index. Explicit formulas are:
\[
\Gut(\SN)=(n-1)(2n-3), \qquad \Gut(\PN)=\frac{1}{3}(n-1)(2n^2-4n+3).
\]
These results can be found in \cite{Das2015, Mukwembi2012} and related works.

Two classical families of graph realizations make this intuition precise.

\begin{definition}[Threshold graph realization~\cite{ChungGrahamSaks1981,Mahadev1995}]
\label{def:threshold}
Given a graphical degree sequence $\mathscr{D}$, the \emph{threshold graph
realization} $\mathrm{TH}(\mathscr{D})$ is the unique (up to isomorphism)
threshold graph with degree sequence $\mathscr{D}$.
\end{definition}
Threshold graphs are constructed greedily, by starting from a single vertex,
each new vertex is added either as an \emph{isolated} vertex (connected to
nobody) or as a \emph{dominating} vertex (connected to all existing vertices),
chosen so that the running degree sequence matches $\mathscr{D}$.

We will also make use of the following well-known tools.

\begin{proposition}[Jensen's Inequality \cite{Demir2025}]\label{pro01jensen}
  If $f\colon I \to \mathbb{R}$ is a convex function on an interval $I$ and
  $x_1,\ldots,x_k \in I$, then
  \[
    f\!\left(\frac{x_1+\cdots+x_k}{k}\right) \;\le\;
    \frac{f(x_1)+\cdots+f(x_k)}{k}.
  \]
  Equality holds if and only if $f$ is affine on $\{x_1,\ldots,x_k\}$ or
  all $x_i$ are equal.
\end{proposition}

\begin{proposition}[Moore Bound \cite{Ibrahim2025REW}]\label{pro01moore}
The maximum number of vertices $b(\Delta, d)$ in a $\Delta$-regular graph of diameter $d$ is given by
\[
b(\Delta, d) =
\begin{cases}
2 & \text{if } \Delta = 1, \\
2d + 1 & \text{if } \Delta = 2, \\
1 + \Delta \dfrac{(\Delta-1)^d - 1}{\Delta-2} & \text{if } \Delta \ge 3.
\end{cases}
\]
\end{proposition}

Since the term $2^{d(u)d(v)}$ grows extremely fast with the product of degrees, the value of $\HGut(G)$ is dominated by edges connecting high-degree vertices. This makes $\HGut$ highly sensitive to the presence of adjacent high-degree vertices, but also causes many non-isomorphic graphs to potentially share the same value when only low-degree edges are present.
Note that calculating $\HGut(G)$ can be done in linear time $O(m)$ once the degree sequence is known.
\begin{observation}
Non-isomorphic graphs can have the same $\HGut$ value.
\end{observation}
For example, the path $\mathcal{P}_4$ and the star $K_{1,3}$ both yield $\HGut = 24$. This shows that $\HGut$ is \emph{not} a complete isomorphism invariant.

A \emph{perfect binary tree} $T_h$ of height $h \geqslant 1$ is defined recursively: $T_0$ consists of a single root vertex. For $h \geq 1$, the root has two children, and each of the two subtrees is a perfect binary tree of height $h-1$. The tree $T_h$ has exactly $n = 2^{h+1}-1$ vertices.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Statement Problem}
The Exponential Gutman index $\HGut(G)$ exhibits interesting behavior due to the rapid growth of the term $2^{d(u)d(v)}$. Several natural questions remain open and deserve further investigation. 
\begin{enumerate}
   \item  
    Determine the maximum and minimum values of $\HGut(G)$ among all connected graphs on $n$ vertices with $m$ edges. For a fixed degree sequence, it is expected that the threshold graph maximizes the index while a suitable bipartite realization minimizes it. Rigorous proofs for general degree sequences are still needed.

    \item The path $\PN$ uniquely minimizes $\HGut(T)$ among all trees on $n$ vertices. Characterize the trees that maximize $\HGut$.
\end{enumerate}
%==================================
\section{Main Results}\label{secmain}
%==================================
Unlike the classical Gutman index, the exponential Gutman index depends only on degree products along edges.
\begin{proposition}~\label{proclas1}
If $G\cong \mathcal{C}_n$. Then, $\HGut(G)=16n$.
\end{proposition}
\begin{proof}
Assume that $G\cong \mathcal{C}_n$ is the cycle graph on $n\geqslant 3$ vertices, then $\HGut(C_n)=16n.$
\end{proof}
In fact, according to Proposition~\ref{proclas1}, if $G\cong \PN$, and $n\geqslant 4$, then $\HGut(G)=16n-40.$ Hence,  $\PN$ has two pendant edges of type $(1,2)$, and $n-3$ internal edges of type $(2,2)$.

\begin{proposition}~\label{proclas2}
Let $a,b \ge 1$ with $a+b = n$.  Then
\begin{equation}~\label{eqq1proclas2}
 \HGut\bigl(\DS(a,b)\bigr) \;=\; 2^{ab} + (a-1)\cdot 2^{a} + (b-1)\cdot 2^{b}.
\end{equation}
\end{proposition}
\begin{proof}
Since $\DS(a,b)$ has one central edge of type $(a,b)$ contributing $2^{ab}$,
then there are $a-1$ pendant edges of type $(a,1)$ each contributing $2^{a}$, and $b-1$ pendant edges of type $(b,1)$ each contributing $2^{b}$. 
\end{proof}

The extremely rapid growth of the function $2^x$ suggests that the value of $\HGut(G)$ contain significant contributions about the multiset of edge degree-products $\mathcal{M}(G) = \{d(u)d(v) \mid uv \in E(G)\}$.

\begin{proposition}~\label{protam2}
Let $G$ and $H$ be two graphs. If $\mathcal{M}(G) \neq \mathcal{M}(H)$. Then, $\HGut(G) = \HGut(H)$. 
\end{proposition}

\begin{proposition}~\label{protam3}
The index $\HGut(G)$ does not uniquely determine the graph $G$ up to isomorphism. In particular, it is not a complete graph invariant.
\end{proposition}


Based on Propositions~\ref{protam2} and \ref{protam3}, we noticed that among graphs with sufficiently large maximum degree product, the exponential terms dominate so strongly that different multisets of edge degree-products are very likely to produce distinct values of $\HGut(G)$. 




\begin{lemma}~\label{lemma01binarytree}
Let $T_h$ be the perfect binary tree of height $h \geqslant 1$ with $n=2^{h+1}-1$ vertices. Then
\begin{equation}~\label{eqq1lemma01binarytree}
\HGut(T_h)=260n-1660
\end{equation}
\end{lemma}
\begin{proof}
If $h=1$, then $T_1$ has a root of degree $2$ connected to two leaves of degree $1$ where $T_1\cong \mathcal{P}_3$. Thus, all edges are of type $(2,1)$, contributing $4$ and according to Proposition~\ref{proclas1},  $\HGut(T_1)=8$. 

In the perfect binary tree with $h \geqslant 2$, the root has degree 2, all non-root internal vertices have degree 3 and 
 there are $2^h$ leaves of degree 1. In this case, we noticed that the edges can be classified into three types as \textit{2 edges} of type (2,3) such that from the root to its two children,
 %and  \textit{$2^{h+1}-4$ edges} of type (3,3) such that between internal vertices. 
and  \textit{$2^h$ edges} of type (3,1) established  from the last internal level to the leaves.
Therefore, at levels $1,\ldots,h-1$: each has degree $3$. Then, there are $2^h-2$ such non-root internal vertices. Thus, the edges between level $\ell$ and level $\ell+1$ for $\ell=1,\ldots,h-2$ are of type $(3,3)$. Hence, there are
      $2^h-4$ such edges. Thus, 
\begin{align*}
  \HGut(T_h)
    &= 2\cdot 2^{2\cdot 3} + (2^h-4)\cdot 2^{3\cdot 3} + 2^h\cdot 2^{3\cdot 1}\\
    &= 2\cdot 2^6 + (2^h-4)\cdot 2^9 + 2^h\cdot 2^3\\
    &= 128 + 512\cdot 2^h - 2048 + 8\cdot 2^h\\
    &= 520\cdot 2^h - 1920.
\end{align*}
Then, by applying value of $n$, the relationship~\eqref{eqq1lemma01binarytree} holds. 
\end{proof}



To determine which trees on a fixed number of vertices minimise and maximise
$\HGut$, we should be move to establishing the most basic property of $\HGut$ among Lemma~\ref{thm01isomorphism}.

\begin{lemma}~\label{thm01isomorphism}
Let $G$ and $H$ be two graphs. If $G \cong H$, then $\HGut(G) = \HGut(H)$.
\end{lemma}
\begin{proof}
Let $ad: V(G) \to V(H)$ be a graph isomorphism. Then $ad$ preserves adjacency, degrees, and distances. In particular, for every edge $uv \in E(G)$, we have $d_G(u) = d_H(ad(u))$, $d_G(v) = d_H(ad(v))$, and $\dist_G(u,v) = \dist_H(ad(u),ad(v)) = 1$. Therefore,
\begin{align*}
 \HGut(G)&= \sum_{uv \in E(G)} \bigl(2^{d_G(u)d_G(v)} + 1\bigr)\\
 &= \sum_{ad(u)ad(v) \in E(H)} \bigl(2^{d_H(ad(u))d_H(ad(v))} + 1\bigr) \\
 &= \HGut(H).   
\end{align*}

Thus, $\HGut$ is an isomorphism invariant.
\end{proof}

Furthermore, by establishing \TGut implies that the converse through Lemma~\ref{thm01isomorphism} is not true. This shows that $\HGut$ is \textit{not} a complete graph invariant.


Actually, by utilizing Proposition~\ref{protam2}, the term $\HGut(G) = \HGut(H)$  holds even when the multisets differ, due to carry-over effects in the base-2 representation. For example, two edges with degree product $p$ contribute $2 \cdot 2^p = 2^{p+1}$, which equals the contribution of a single edge with degree product $p+1$.
Nevertheless, since the term $2^{d(u)d(v)}$ grows so quickly, collisions become relatively rare for graphs with high maximum degree or large number of edges.

We now investigate extremal properties of $\HGut$ among trees.

\begin{theorem}~\label{thm001extremal}
Among all trees on $n$ vertices, the path $\PN$ is a strong candidate for the minimum value of $\HGut$. 
\end{theorem}
\begin{proof}
The path graph contains only edges of types $(1,2)$ and $(2,2)$, whose contributions are respectively $4$ and $16$. 
Suppose that $T$ is a tree on $n$ vertices that is not a path. Then $T$ contains at least one vertex of degree at least $3$. Consequently, at least one edge $uv$ satisfies $d_u\,d_v\geqslant 3$. 
Assume the result holds for every tree on $n$ vertices where $n\geqslant 5$.  Let $T$ be
an arbitrary tree on $n$ vertices.  Choose any leaf $u$ of $T$ and let
$v= N_T(u)$ be its unique neighbour. Another tree derived from $T$ as $T'=T-u$, the tree on $n-1$
vertices obtained by deleting $u$ and the edge $uv$. In this case, suppose that $k=d_T(v)$ be the degree of $v$ in $T$. Then, $d_{T'}(v)=k-1$, and the neighbours of $v$ in $T$ other than $u$ are $w_1,\ldots,w_{k-1}$, with
degrees $d_{w_1},\ldots,d_{w_{k-1}}$ in $T$, it remains unchanged in $T'$.
Thus, we should be comparing between $\HGut(T)$ and $\HGut(T')$. For satisfies this purpose, we noticed that the edge $uv$ is present in $T$ but not in $T'$; it contributes $2^{k\cdot 1}=2^k$. Thus, each edge $vw_i$ changes its contribution from $2^{k\cdot d_{w_i}}$ in $T$
        to $2^{(k-1)d_{w_i}}$ in $T'$.
Therefore,
\begin{equation}~\label{eqq1thm001extremal}
  \HGut(T) = \HGut(T') + 2^{k}
  + \sum_{i=1}^{k-1}\bigl[2^{k\,d_{w_i}} - 2^{(k-1)d_{w_i}}\bigr].
\end{equation}
Thus, according to Proposition~\ref{proclas1}, if $G\cong \PN$, and $n\geqslant 4$, $\HGut(T') \geqslant \HGut(\mathcal{P}_{n-1})$, and $\HGut(\mathcal{P}_{n-1})=16n-56$. Hence, 
\begin{equation}~\label{eqq2thm001extremal}
  \Delta_k:= 2^{k} + \sum_{i=1}^{k-1} 2^{(k-1)d_{w_i}}(2^{d_{w_i}}-1),
\end{equation}
where $ \Delta_k\geqslant 16$. 


\noindent\textbf{Case $k=2$.}
There is exactly one neighbour $w_1$ of $v$ in $T'$.
Since $T'$ is a connected tree on $n\geqslant 5$ vertices with $v$ a leaf of degree $k$), the vertex $w_1$ must have degree $d_{w_1} \geqslant 2$ in $T'$ equivalently, in $T$.
Then, $\Delta_2= 4 + 2^{d_{w_1}}(2^{d_{w_1}}-1)$ it implies that
\[
  \Delta_2=4+ 2^{2d_{w_1}}-2^{d_{w_1}}.
\]
Since $d_{w_1}\geqslant 2$, we have $2^{2d_{w_1}}-2^{d_{w_1}} \geqslant  12$. Thus, $\Delta_2 \geqslant 16$, with equality if and only if $d_{w_1}=2$.

\noindent\textbf{Case $k=3$.}
We have two neighbours $w_1, w_2$ of $v$ in $T'$.  Since $n \geqslant 5$, not both
can be leaves in $T'$; otherwise $T$ would have $n \leqslant 4$ vertices,
contradicting $n \geqslant 5$ with $T' = K_{1,2}$); at least one, say $w_2$, satisfies
$d_{w_2} \ge 2$.  If $d_{w_1} = 1$, $\Delta_3 = 8 + 4(2^1-1) + 2^{2d_{w_2}}(2^{d_{w_2}}-1)$. Thus, 
\[
  \Delta_3= 8 + 4 + (2^{2d_{w_2}}-2^{d_{w_2}}), 
\]
where $\Delta_3 \geqslant 16$. If $d_{w_1} \geqslant 2$ also, 
$\Delta_3 \geqslant 16$.

If $k \geqslant 4$. We have $2^k \geqslant 16$ and the sum is non-negative, so $\Delta_k \geqslant 16$.
Thus, in all cases $\Delta_k \geqslant 16$, establishing \eqref{eqq2thm001extremal} and hence
$\HGut(T) \geqslant \HGut(\mathcal{P}_{n-1})+16=\HGut(\PN)$.

Moreover, the presence of a branching vertex increases the degree products of adjacent edges compared with the path structure. Since the exponential function is strictly increasing, every such increase raises the total value of the sum $\HGut$.
\end{proof}

\begin{lemma}~\label{lemma02edgebound}
Let $T$ be a tree on $n$ vertices and let $uv \in E(T)$.
  Removing $uv$ from $T$ yields two components of sizes $t$ and $n-t$.
  Then
\[
d_u \cdot d_v \;\leqslant\; t(n-t) \;\leqslant\;
    \left\lfloor\frac{n}{2}\right\rfloor\!\left\lceil\frac{n}{2}\right\rceil.
\]
Equality $d_u d_v = \lfloor n/2\rfloor\lceil n/2\rceil$ holds if and only if  $t= \lfloor n/2\rfloor$, $d_u=t$, $d_v=n-t$, where $T =\DS\,\bigl(\lfloor n/2\rfloor, \lceil n/2\rceil\bigr)$   and $uv$ is its central edge.
\end{lemma}
\begin{proof}
In the component of size $s$ containing $u$, the vertex $u$ is adjacent to
at most $t-1$ other vertices in that component with $v$. Thus, $d_u\leqslant t$. Since $d_v \leqslant n-t$ yields  $d_u\,.\,d_v \leqslant t(n-t)$. In this case, the term $t(n-t)$ is maximised over integers $1 \leqslant t \leqslant n-1$ where $t=\lfloor n/2\rfloor$, it implies that $t(n-t)=\lfloor n/2\rfloor\lceil n/2\rceil$.  Hence, for the equality condition, $d_u=t$ needs the vertex $u$ to be connected to all
$t-1$ other vertices in its component, so that component is a star with
centre $u$ and $t-1$ leaves.  Similarly, if $d_v=n-t$ forces the other
component to be a star with centre $v$ and $n-t-1$ leaves.
Therefore $T=\DS(t,n-t)=\DS\,\bigl(\lfloor n/2\rfloor,\lceil n/2\rceil\bigr)$.
\end{proof}

Based on Lemma~\ref{lemma02edgebound}, we have $\HGut\!\bigl(\DS(a,b)\bigr)=2^{ab} + (a-1)\cdot 2^{a} + (b-1)\cdot 2^{b},$ which it will utilize in next theorem.

\begin{theorem}~\label{Thm001max}
Let $a=\lfloor n/2\rfloor$ and $b=\lceil n/2\rceil$. Then, among all trees on $n \geqslant 4$ vertices,
\begin{equation}~\label{eqq1Thm001max}
   \HGut(T) \;\leqslant\; \HGut\!\bigl(\DS(a,b)\bigr)
\end{equation}
with equality if and only if $T=\DS(a,b)$, for $n \geqslant 5$ or
$T \in \{P_4, K_{1,3}\}$, for $n=4$.
\end{theorem}
\begin{proof}
By Lemma~\ref{lemma02edgebound}, every edge $uv$ of $T$ satisfies
$d_u d_v \leqslant ab$. The,  $2^{d_u d_v} \leqslant 2^{ab}$. Thus, if $T=\DS(a,b)$, then $\HGut(T)=2^{ab}+(a-1)2^a+(b-1)2^b$ where it established among Proposition~\ref{proclas2}.

\case{1} For $n=4$, we noticed that both $\mathcal{P}_4=\DS(2,2)$ and $K_{1,3}=\DS(1,3)$, and there are no other trees on $4$ vertices.

\case{2} For $n \geqslant 5$, we find that $\HGut(\DS(a,b)) > \HGut(T)$ for any $T\ncong \DS(a,b)$. Thus, by utilizing  Lemma~\ref{lemma02edgebound}, every edge of $T$ has degree product at most
$ab-1$. Since the maximum $ab$ is achieved only at the central edge of $\DS(a,b)$.
Thus
\[
  \HGut(T) \;\leqslant\; (n-1)\cdot 2^{ab-1}.
\]
\noindent \textbf{Claim 1.} $\HGut(\DS(a,b)) > (n-1)\cdot 2^{ab-1}$ for $n \geqslant 5$.

\medskip 

Since $2^{ab}>(n-1)\cdot 2^{ab-1}$ where  $2 > n-1$, and  $n < 3$, this
crude bound is not sufficient on its own.  We instead compare
$\DS(a,b)$ directly with the double star
$\DS(a-1,b+1)$, which by Lemma~\ref{lemma02edgebound} has the largest
edge degree product among all trees other than $\DS(a,b)$,
\begin{align*}
  &\HGut(\DS(a,b))-\HGut(\DS(a{-}1,b{+}1)) \\
  &\quad = \bigl[2^{ab}-2^{(a-1)(b+1)}\bigr]
         + \bigl[a\cdot 2^{a-1}\bigr]
         - \bigl[(b+1)\cdot 2^{b}\bigr].
\end{align*}
Since $b \geqslant a$ implies $(a-1)(b+1) = ab+(a-b-1) \leqslant ab-1$, we have
\[
  2^{ab}-2^{(a-1)(b+1)} \;=\; 2^{(a-1)(b+1)}\bigl(2^{b-a+1}-1\bigr)
  \;\geqslant\; 2^{(a-1)(b+1)}.
\]
Thus, by considering $a \geqslant 2$ and $b \geqslant 3$  where $n \geqslant 5$, $ab \geqslant 6$ and
$(a-1)(b+1) \geqslant 4$. Then, $2^{(a-1)(b+1)} \geqslant 16$. Thus, 
$2^{(a-1)(b+1)}(2^{b-a+1}-1)>(b+1)\cdot 2^b-a\cdot 2^{a-1}$
holds for all $a,b$ with $a+b=n \geqslant 5$ and $a \leqslant b$.
\hfill $\blacklozenge$
%

Therefore, for any non-$\DS(a,b)$ tree $T$ has maximum edge degree product at most
$(a-1)(b+1)$, then the product for $\DS(a-1,b+1)$. Then, according to case 1, case 2 and Claim 1,  we have
$\HGut(T) \leqslant \HGut(\DS(a-1,b+1)) < \HGut(\DS(a,b))$, which completing the proof.
\end{proof}
To establish the maximum term it needs to verified value in Theorem~\ref{Thm001max}, for $n = 5,6,7$ by direct computation in Table~\ref{tab001HGutSmall}. Hence,  it emphasizes that the star graph does not maximize $\HGut$ among trees.

\begin{table}[H]
  \centering
  \renewcommand{\arraystretch}{1.3}
  \begin{tabular}{l|l|c|c}
    \hline
    $n$ & Tree $T$ & Degree sequence & $\HGut(T)$ \\
    \hline
    5 & $\mathcal{P}_5$ & $(1,2,2,2,1)$ & $40$ \\ \hline
    5 & $K_{1,4}$ & $(4,1,1,1,1)$ & $64$ \\ \hline
    5 & $\DS(2,3)$ & $(3,2,1,1,1)$ & $84$ \\ 
    \hline
    6 & $P_6$ & $(1,2,2,2,2,1)$ & $56$ \\ \hline
    6 & ``Broom'' & $(1,2,2,3,1,1)$ & $100$ \\ \hline
    6 & ``Spider'' & $(3,2,2,1,1,1)$ & $144$ \\ \hline
    6 & $K_{1,5}$ & $(5,1,1,1,1,1)$ & $160$ \\ \hline
    6 & $\DS(2,4)$ & $(4,2,1,1,1,1)$ & $308$ \\ \hline
    6 & $\DS(3,3)$ & $(3,3,1,1,1,1)$ & $544$ \\
    \hline
  \end{tabular}
   \caption{All non-isomorphic trees on $n=5,6$ vertices with their
           $\HGut$ values, confirming that $\DS\!\bigl(\lfloor n/2\rfloor,\lceil n/2\rceil\bigr)$ is the unique maximum and $\PN$ is the unique minimum.}
  \label{tab001HGutSmall}
\end{table}

 Indeed, for $n \geqslant 5$, we find that $ \HGut(K_{1,n-1}) = (n-1)\cdot 2^{n-1},$
  where it bounded by $\HGut\!\bigl(\DS(a,b)\bigr) \geqslant 2^{ab} = 2^{\lfloor n^2/4\rfloor},$
  and $\lfloor n^2/4\rfloor > n-1$ for all $n \ge 5$, so
  $2^{ab} \gg (n-1)\cdot 2^{n-1}$.
  For example, according to Table~\ref{tab001HGutSmall},  at $n=6$, $\HGut(K_{1,5}) = 160$ while
  $\HGut(\DS(3,3)) = 544$.
  This contrasts sharply with the classical Gutman index, where $K_{1,n-1}$
  is the minimum and $\PN$ is the maximum.
  The exponential amplification of degree products fundamentally changes the
  extremal structure.



%-------------------------------------------------------------
\subsection{Bounds Among Classical Invariants}~\label{subsecbounds}
%-------------------------------------------------------------
In this subsection, we began with lower bound among Jensen's inequality by Lemma~\ref{lemlowergut01} and the upper bound among the Moore bound had established  by Lemma~\ref{lemuppergut02}.

\begin{lemma}~\label{lemlowergut01}
  Let $G$ be a graph with $m\geqslant 1$ edges and let
  $M_2(G)$ be the second Zagreb index. Then
\begin{equation}\label{eqq1lemlowergut01}
\HGut(G) \;\geqslant\; m\cdot 2^{\,M_2(G)/m},
\end{equation}
with equality if and only if $d_u\, d_v$ is constant over all edges $uv \in E(G)$.
\end{lemma}
\begin{proof}
  The function $f(x)=2^x$ is strictly convex on $\mathbb{R}$.
  Applying Proposition~\ref{pro01jensen} to the $m$ values $\lambda_{uv} = d_u d_v$
  gives
  \[
  \frac{1}{m}\sum_{uv\in E(G)} 2^{\lambda_{uv}}
    \;\geqslant\; 2^{\frac{1}{m}\sum_{uv\in E(G)}\lambda_{uv}}
    =2^{M_2(G)/m}.
  \]
  By using transform as multiply both sides by $m$ yields \eqref{eqq1lemlowergut01}.
  Since $f$ is strictly convex, equality holds if and only if all $d_u d_v$
  are equal. In this case, the graph is \emph{edge-regular} with respect to degree products.  This holds in particular for every regular graph and every biregular bipartite graph.
\end{proof}

\begin{lemma}~\label{lemuppergut02}
  Let $G$ be a connected graph with maximum degree $\Delta \ge 1$ and diameter
  $d \ge 1$.  Then, according to Proposition~\ref{pro01moore}, 
\begin{equation}\label{eqq1lemuppergut02}
\HGut(G) \;\leqslant\; 2^{\Delta^2}\cdot\frac{\Delta}{2}\cdot b(\Delta,d).
\end{equation}
\end{lemma}
\begin{proof}
  For every edge $uv$ one has $d_u\le\Delta$ and $d_v\le\Delta$, so
  $2^{d_u d_v}\le 2^{\Delta^2}$.
  Hence $$\HGut(G) \le m\cdot 2^{\Delta^2}.$$
  Since $G$ has maximum degree $\Delta$ and diameter $d$, according to Proposition~\ref{pro01moore}, the Moore bound gives
  $n \leqslant b(\Delta,d)$. Thus,  $2m \leqslant \Delta\, n \leqslant \Delta\, b(\Delta,d)$. Thus, we have $m \leqslant \tfrac{\Delta}{2}\,b(\Delta,d)$. Thus, the relationship~\eqref{eqq1lemuppergut02} holds. 
\end{proof}

The upper bound in Lemma~\ref{lemuppergut02} is approached (though not attained for most parameters) only when $G$ is close to a Moore graph, where $\Delta$-regular is formed of diameter $d$, with $n = b(\Delta,d)$ vertices. 
Known Moore graphs include complete graphs $K_{\Delta+1}$ ($d=1$), and odd cycles where $\Delta=2$.

Actually, together Lemmas~\ref{lemlowergut01} and~\ref{lemuppergut02} provide significantly sharper bounds than the naive estimates   $2m\leqslant \HGut(G) \leqslant m\cdot 2^{\Delta^2}$, because the lower bound is tight for regular graphs and the upper bound incorporates structural constraints.

\subsection{Extremal Values for Fixed Degree Sequences}
\label{subsecextremal}
In this subsection, consider $G$ be a simple connected graph on $n$ vertices with degree sequence
$\mathscr{D} = (d_1 \ge d_2 \ge \cdots \ge d_n)$.
We study which graph realizations of $\mathscr{D}$ attain the maximum or
minimum value of
\[
   \SG(G)\;:=\; \sum_{uv \in E(G)} 2^{d_u d_v},
\]
which is the combinatorially significant part of $\HGut(G)$.
Intuitively, $\SG(G)$ is large when edges connect vertices of large degree to each other, and small when large degree vertices are matched to small degree vertices.

\begin{proposition}
\label{prop001thresholdmax}
Let $\mathscr{D}$ be a graphical degree sequence.
Among all connected realizations $G$ of $\mathscr{D}$, the threshold graph
realization $\mathrm{TH}(\mathscr{D})$ attains the maximum value of $\SG(G)$.
\end{proposition}
\begin{proof}
We use a swap argument based on the rearrangement inequality for convex functions (see, e.g., \cite{Marshall2011Inequalities}).
Suppose $G$ is a connected realization of $\mathscr{D}$ that is \emph{not}
a threshold graph.
Then there exist four distinct vertices $a, b, c, d$ with
$d_a \ge d_b$, $d_c \ge d_d$, such that $ac \in E(G)$, $bd \in E(G)$,
but $bc \notin E(G)$ and $ad \notin E(G)$
(this configuration characterizes non-threshold graphs; see
\cite{ChungGrahamSaks1981}).
Consider the \emph{2-switch} that replaces edges $\{ac, bd\}$ with
$\{bc, ad\}$ (or $\{ab, cd\}$, depending on the configuration); this
operation preserves the degree sequence.
Since $d_a \ge d_b$ and $d_c \ge d_d$, we have $d_a d_c + d_b d_d \;\geqslant \; d_a d_d + d_b d_c.$ Then, 
\[
   2^{d_a d_c} + 2^{d_b d_d} \;\geqslant\; 2^{d_a d_d} + 2^{d_b d_c}.
\]
Hence the 2-switch from $\{bc, ad\}$ to $\{ac, bd\}$ does not decrease $\SG(G)$.
Thus, according to~\cite{ChungGrahamSaks1981},
we conclude that $\mathrm{TH}(\mathscr{D})$ is a local
maximum of $\SG(G)$ over all realizations of $\mathscr{D}$.
\end{proof}

The 2-switch argument among Proposition~\ref{prop001thresholdmax} shows that \emph{some} threshold realization
maximizes $\SG(G)$.
Since $\mathscr{D}$ has a unique threshold realization which occurs for most sequences, the maximizer is unique.
For degree sequences with multiple threshold realizations, all of them attain the same value of $\SG(G)$.

For the minimum, the situation is more subtle such that not every degree sequence has a bipartite realization. Thus, we state the result conditionally.

\begin{proposition}
\label{prop001bipartitemin}
Let $\mathscr{D}$ be a graphical degree sequence.
\begin{enumerate} 
  \item If $\mathscr{D}$ has a bipartite realization $\mathrm{BR}(\mathscr{D})$,
        then among all connected bipartite realizations of $\mathscr{D}$, those
        in which high-degree vertices are in opposite parts of the bipartition
        minimize $S(G)$.
  \item Among \emph{all} connected realizations of $\mathscr{D}$, a minimizer
        exists but need not be bipartite; it is, however, a graph in which the
        edges of largest degree-product are avoided as much as possible.
\end{enumerate}
\end{proposition}
\begin{proof} According to Proposition~\ref{prop001thresholdmax}, we will utilize it for same 2-switch argument applied in reverse.

\noindent \textbf{(1).} In this case, by replacing
an edge $ac$ with $a, c$ both of large degree, with an edge
connecting a large-degree vertex to a small-degree vertex decreases
$2^{d_a d_c}$ by strict convexity. Hence, we find that the largest degree vertices in each part are
\emph{not} adjacent minimizes  $\SG(G)$. 

\noindent \textbf{(2).} Since $\SG(G)$ is a continuous function of a discrete edge set. Then, a minimum over the finite set of realizations always exists. Thus, 
large degree product edges follows from the same convexity argument.
\end{proof}

A fully rigorous treatment of both extremal problems for all degree sequences requires a careful application of the \emph{rearrangement inequality} or \emph{majorization} (in the sense of Hardy–Littlewood–P\'{o}lya; see
\cite{Marshall2011Inequalities, HardyLittlewoodPolya1952}).
In particular, one needs to verify that the sequence of degree products
$(d_u d_v)_{uv \in E(G)}$ is majorized by the corresponding sequence for
$\mathrm{TH}(\mathscr{D})$, which would then imply $S(G) \le S(\mathrm{TH}(\mathscr{D}))$
for all convex $f$ by the Schur-convexity of the symmetric sum.
We leave the complete majorization proof as a direction for future work.


\subsection{On the Inverse Problem and Uniqueness}
\label{sbsecinverse}
Given a positive integer
$\kappa$, determine all graphs $G$ up to isomorphism satisfying
$\HGut(G)=\kappa$.
This is a natural question for any graph index.

\medskip 

We first show that $\HGut$ is \emph{not} a complete graph invariant, i.e., it
does not distinguish all non-isomorphic graphs. For example, $\SG(\mathcal{P}_4)=\SG(K_{1,3})=24$ even though $\mathcal{P}_4\not\cong K_{1,3}$. 
Despite the above counterexample, collisions appear to become increasingly rare as the value of the index grows.
The reason is the extremely rapid growth of $2^{d_u d_v}$ as a single edge between two vertices of degree $k$
contributes $2^{k^2}$, which dominates the entire sum for all smaller edges.

Assume that $d_u d_v=\lambda_{uv}$ and all other edges
$ab$ satisfy $d_a\,d_b<\lambda_{uv}$. Then, 
\[
   \SG(G) \;=\; 2^{\lambda_{uv}}+ R(G),
\]
where $R(G) < (m-1)\cdot 2^{\lambda_{uv}-1}.$ Thus,  the term $2^{\lambda_{uv}}$ determines a narrow range in which any other graph
$H$ with $\SG(H)=\SG(G)$ should be also have its maximum edge degree product equal to $\lambda_{uv}$.
Thus, this severely constrains the structure of graphs that can produce the same index value.

We note that Conjecture~\ref{conj01unique} is analogous to known
results for other graph polynomials and indices: for example, almost all trees are determined by their chromatic polynomial \cite{Read1968}, and similar almost all graphs are determined by their spectrum results hold for the
adjacency matrix \cite{vanDamHaemers2003}.
Whether $\HGut$ behaves similarly is an interesting open question.

%\section{Open Problems}
Characterizing all values $\kappa$ that are attained by some graph, and all graphs that attain a given value, remains open.
We record two natural questions.

\begin{question}~\label{q1inverse1}
For which positive integers $\kappa$ does there exist a graph $G$ with
$\HGut(G) = \kappa$?
\end{question}
Equivalently based on Question~\ref{q1inverse1}, what is the image of $\HGut$ over the class of all connected graphs?
\begin{question}~\label{q2inverse2}
Is the set $\{G : \HGut(G) = \kappa\}$ finite for every $\kappa$?
For fixed $n$, is there an effective algorithm to determine all $n$-vertex
graphs with a prescribed value of $\HGut$?
\end{question}

To determine behavior under graph operations, we must study how $\HGut(G)$ changes under standard graph operations.
 We also propose the following conjecture concerning eventual uniqueness.

\begin{conjecture}~\label{conj01unique}
There exists a constant $\kappa_0$ such that depending only on the class of graphs
under consideration, e.g., trees on $n\geqslant 3$ vertices. 
\end{conjecture}
Actually, according to Conjecture~\ref{conj01unique} and based on~\ref{q2inverse2},  for all
$\kappa \geqslant \kappa_0$, there is at most one graph $G$ (up to isomorphism) with $\HGut(G) = \kappa$. 
The heuristic support for Conjecture~\ref{conj01unique} is the
following.
For $\kappa$ to be achieved by two non-isomorphic graphs $G$ and $H$, their
multisets of edge degree products $\{\lambda_{uv}: uv \in E\}$ must produce equal $\SG(G)$.
Since the terms $2^{\lambda_{uv}}$ grow doubly exponentially in the maximum degree, distinct multisets of degree products generically yield distinct sums of powers
of two; collisions require a precise cancellation that becomes increasingly unlikely as $\kappa$ increases.



\section{Conclusion}

We have studied the exponential Gutman index
$\HGut(G)$ for trees. In this paper,  we emphasize that the path graph $\PN$ minimises $\HGut$ among all trees on $n \geqslant 4$ vertices, with $\HGut(\PN)=16n-40$. Also, the balanced double star
$\DS(\lfloor n/2\rfloor,\lceil n/2\rceil)$ maximises $\HGut$ for $n\geqslant 5$, among the sharp edge degree product bound of Lemma~\ref{lemma02edgebound}. In particular, the star $K_{1,n-1}$ does \emph{not} maximise $\HGut$; it is far from optimal because the exponential amplification favours adjacent high degree vertices over the star's hub leaf structure.
Through Lemma~\ref{lemma01binarytree},  for complete binary trees, showing $\HGut(T_h)=260n-1660$ for height $h\geqslant 2$.
The bounds of $\HGut$ had established  as lower bound by utilizing Jensen's inequality amng Lemma~\ref{lemlowergut01}) and an upper
bound by utilizing the Moore bound among Lemma~\ref{lemuppergut02}. 

These results and formulas support efficient computation and deeper analysis of distance-degree invariants, enabling future studies on optimal degree arrangements in general graphs.

%===========================
\section*{Declarations}
\begin{itemize}
	\item Funding: Not Funding.
	\item Conflict of interest/Competing interests: The author declare that there are no conflicts of interest or competing interests related to this study.
	\item Ethics approval and consent to participate: The author contributed equally to this work.
	\item Data availability statement: All data is included within the manuscript.
\end{itemize}

\bigskip
\begin{thebibliography}{1} % Ññûëêè íà ëèòåðàòóðó ïðèâîäÿòñÿ òîëüêî íà àíãëèéñêîì ÿçûêå (äàæå åñëè ó èñòî÷íèêà íåò àíãëîÿçû÷íîé âåðñèè).


\bibitem{Andova2012}
V.~Andova, D.~Dimitrov, J.~Fink, R.~\v{S}krekovski,
Bounds on Gutman index,
\textit{MATCH Commun. Math. Comput. Chem.} \textbf{67} (2012), 512--524.

\bibitem{ChungGrahamSaks1981}
F.~R.~K.~Chung, R.~L.~Graham, M.~E.~Saks,
A dynamic location problem for graphs,
\textit{Combinatorica} \textbf{9} (1989), 111--131.

\bibitem{Das2015}
K.~Ch.~Das, M.~J.~Nadjafi-Arani,
Relations between distance-based and degree-based topological indices,
\textit{Appl. Math. Comput.} \textbf{270} (2015), 142--147.

\bibitem{Demir2025}
I.~Demir,
Refinements of the Jensen Inequality and Estimates of the Jensen Gap Based on Interval-Valued Functions,
{\em Mathematical Methods in the Applied Sciences} (2025).

\bibitem{reef1}
D.~Bollob{\'a}s,
The Wiener number---some applications and new developments,
{\it Discrete Math.} \textbf{91} (1991), 58--85.

\bibitem{reef4}
A.~Dobrynin, R.~Entringer, I.~Gutman,
Wiener index of trees: theory and applications,
{\it Acta Appl. Math.} \textbf{66} (2001), 211--249.

\bibitem{GutIndex}
I.~Gutman,
Selected properties of the Schultz molecular topological index,
{\em J. Chem. Inf. Comput. Sci.} \textbf{34} (1994), 1087--1089.

\bibitem{reef10}
I.~Gutman, O.~Ivanov, E.~Polansky,
{\it Mathematical Concepts in Organic Chemistry},
Springer, Berlin, 1986.

\bibitem{Gutman1973}
I.~Gutman, N.~Trinajsti\'c,
Graph theory and molecular orbitals. Total $\pi$-electron energy of alternant hydrocarbons,
{\em Chem. Phys. Lett.} \textbf{17} (1972), 535--538.

\bibitem{new3}
I.~Gutman, B.~Ru\v{s}\v{c}i\'c, N.~Trinajsti\'c, C.~F.~Wilcox,
Graph theory and molecular orbitals. XII. Acyclic polyenes,
{\em Journal of Chemical Physics} \textbf{62} (1975), 3399--3405.

\bibitem{HardyLittlewoodPolya1952}
G.~H.~Hardy, J.~E.~Littlewood, G.~P\'olya,
\textit{Inequalities}, 2nd edition, Cambridge University Press, 1952.

\bibitem{Ibrahim2025REW}
R.~Ibrahim, H.~LaFayette, K.~McCall,
Minimum 2-percolating sets in 2-connected, diameter 2 graphs,
{\em Australasian Journal of Combinatorics} \textbf{93}(1) (2025), 60--89.

\bibitem{Mahadev1995}
N.~V.~R.~Mahadev, U.~N.~Peled,
Threshold Graphs and Related Topics,
Annals of Discrete Mathematics, 56, North-Holland, Amsterdam, 1995.

\bibitem{Marshall2011Inequalities}
A.~W.~Marshall, I.~Olkin, B.~C.~Arnold,
\textit{Inequalities: Theory of Majorization and Its Applications}, 2nd edition, Springer, 2011.

\bibitem{Mazorodze2014}
J.~P.~Mazorodze, S.~Mukwembi, T.~Vetr{\'i}k,
On the Gutman index and minimum degree,
{\em Discrete Applied Mathematics} \textbf{173} (2014), 77--82.

\bibitem{Mukwembi2012}
S.~Mukwembi,
On the upper bound of Gutman index of graphs,
\textit{MATCH Commun. Math. Comput. Chem.} \textbf{68} (2012), 343--348.

\bibitem{Read1968}
R.~C.~Read,
An introduction to chromatic polynomials,
\textit{Journal of Combinatorial Theory} \textbf{4} (1968), 52--71.

\bibitem{Schultz1989}
H.~P.~Schultz,
Topological organic chemistry. 1. Graph theory and topological indices of alkanes,
{\em J. Chem. Inf. Comput. Sci.} \textbf{29} (1989), 227--228.

\bibitem{Todeschini2010}
R.~Todeschini, V.~Consonni,
{\em Handbook of Molecular Descriptors},
Wiley-VCH, Weinheim, 2010.

\bibitem{vanDamHaemers2003}
E.~R.~van~Dam, W.~H.~Haemers,
Which graphs are determined by their spectrum?,
\textit{Linear Algebra and Its Applications} \textbf{373} (2003), 241--272.

\bibitem{Wiener1947}
H.~Wiener,
Structural determination of paraffin boiling points,
{\em J. Am. Chem. Soc.} \textbf{69} (1947), 17--20.
\end{thebibliography}
\end{document}
