On the Estrada Indices of Unicyclic Graphs with Fixed Diameters

Wenjie Ning,Kun Wang

doi:10.3390/math9192395

Abstract

The Estrada index of a graph G is defined as EE(G)=∑i=1neλi, where λ1,λ2,…,λn are the eigenvalues of the adjacency matrix of G. A unicyclic graph is a connected graph with a unique cycle. Let U(n,d) be the set of all unicyclic graphs with n vertices and diameter d. In this paper, we give some transformations which can be used to compare the Estrada indices of two graphs. Using these transformations, we determine the graphs with the maximum Estrada indices among U(n,d). We characterize two candidate graphs with the maximum Estrada index if d is odd and three candidate graphs with the maximum Estrada index if d is even.

Highlights

We show two candidate graphs with the maximal Estrada index if d is odd and three candidate graphs with the maximal Estrada index if d is even
We propose a hypothesis on the structure of the extremal graph with the maximum Estrada index in U (n, d)
By Claims 5 and 3, if l = k + 1, G is the unicyclic graph with the maximum Estrada index of diameter d obtained from Pd and vd+1 by adding the edges vk vd+1 and vk+2 vd+1, and attaching n − d − 2 pendant edges at one vertex v ∈ V ( P) for some 1 ≤ k ≤ d − 2

Summary

Introduction

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations This graph invariant was first proposed as a measure of the degree of folding of a protein [1] and has been found multiple applications in various fields, such as measurements of the subgraph centrality and the centrality of complex networks [2,3] and the extended molecular branching [4]. Many results have been obtained on characterizing graphs that maximize (or minimize) the Estrada index among a given set of graphs. Denote by U (n, d) the set of all unicyclic graphs with n vertices and diameter d.

Preliminaries

Lemmas

Conclusions

Methods