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New bounds of extended energy of a graph

Abujafar Mandal, Sk. Md. Abu Nayeem

TL;DR

This work introduces the extended vertex energy via the extended adjacency matrix $A_{ex}$ and derives both upper and lower bounds for the extended energy $\varepsilon_{ex}(G)$ of graphs. It presents two new vertex-energy bounds, leading to two novel global bounds on $\varepsilon_{ex}(G)$ that tighten existing results and show improvements in several graph families. The paper also develops Nordhaus–Gaddum-type bounds for $\varepsilon_{ex}(G)$ and $\varepsilon_{ex}(\bar{G})$, with stronger inequalities under natural connectivity assumptions. Overall, the results enhance the spectral-graph inequalities landscape by linking $\varepsilon_{ex}$ to order, size, degree extrema, and structural configurations, and by providing explicit equality cases. These bounds have potential implications for graph energy theory and related applications in chemistry and network analysis.

Abstract

The extended adjacency matrix of a graph with $n$ vertices is a real symmetric matrix of order $n\times n$ whose $(i,j)$-th entry is the average of the ratio of the degree of the vertex $i$ to that of the vertex $j$ and its reciprocal when $i,j$ are adjacent and zero otherwise. The aggregate of absolute eigenvalues of the extended adjacency matrix is termed the extended energy. In this paper, the concept of extended vertex energy is introduced, and some bounds of extended vertex energy are obtained. From there, we establish some new upper bounds of the extended energy of a graph involving order, size, largest, and smallest degree. We show that those are improvements of some existing bounds. Through direct manipulation, we have also established some more upper and lower bounds of extended energy, which are either better or incomparable with the existing bounds. Finally, some improved bounds of Nordhaus-Gaddum-type are found.

New bounds of extended energy of a graph

TL;DR

This work introduces the extended vertex energy via the extended adjacency matrix and derives both upper and lower bounds for the extended energy of graphs. It presents two new vertex-energy bounds, leading to two novel global bounds on that tighten existing results and show improvements in several graph families. The paper also develops Nordhaus–Gaddum-type bounds for and , with stronger inequalities under natural connectivity assumptions. Overall, the results enhance the spectral-graph inequalities landscape by linking to order, size, degree extrema, and structural configurations, and by providing explicit equality cases. These bounds have potential implications for graph energy theory and related applications in chemistry and network analysis.

Abstract

The extended adjacency matrix of a graph with vertices is a real symmetric matrix of order whose -th entry is the average of the ratio of the degree of the vertex to that of the vertex and its reciprocal when are adjacent and zero otherwise. The aggregate of absolute eigenvalues of the extended adjacency matrix is termed the extended energy. In this paper, the concept of extended vertex energy is introduced, and some bounds of extended vertex energy are obtained. From there, we establish some new upper bounds of the extended energy of a graph involving order, size, largest, and smallest degree. We show that those are improvements of some existing bounds. Through direct manipulation, we have also established some more upper and lower bounds of extended energy, which are either better or incomparable with the existing bounds. Finally, some improved bounds of Nordhaus-Gaddum-type are found.
Paper Structure (7 sections, 22 theorems, 58 equations)

This paper contains 7 sections, 22 theorems, 58 equations.

Table of Contents

  1. Introduction
  2. Extended energy of a vertex and its bounds
  3. Upper bounds of extended vertex energy
  4. A lower bound of extended vertex energy
  5. Upper bounds of extended energy
  6. Lower bound of extended energy
  7. Nordhaus-Gaddum-type bounds of extended energy

Key Result

Lemma 1

Let $G=(V,E)$ be an $n$ vertices graph where the vertices are labelled as $1,2,\ldots,n$. Then for each $i\in V(G)$ where $\eta_r$ is the $r$-th eigenvalue of $A_{ex}$, the extended adjacency matrix of the graph $G$ and the weights $q_{ir}$ satisfy Moreover, $q_{ir}=w_{ir}^2$ where $W=(w_{ir})$ is an orthogonal matrix formed by the eigenvectors of $A_{ex}$ as column vectors.

Theorems & Definitions (32)

  • Definition 1
  • Lemma 1
  • Lemma 2: McClelland
  • Lemma 3
  • Theorem 1
  • Theorem 2
  • Lemma 4
  • Theorem 3
  • Lemma 5
  • Lemma 6
  • ...and 22 more