The Signless Laplacian Estrada Index of Unicyclic Graphs

TL;DR

This paper investigates the signless Laplacian Estrada index (SLEE) of unicyclic graphs, characterizing extremal graphs and identifying the maximum SLEE graph for given diameter.

Contribution

It characterizes unicyclic graphs with extremal SLEE values and determines the unique maximum SLEE unicyclic graph for fixed diameter.

Findings

Identified unicyclic graphs with the top two largest and smallest SLEE values.

Determined the unique unicyclic graph with maximum SLEE for a given number of vertices and diameter.

Abstract

For a graph $G$, the signless Laplacian Estrada index is defined as $SLEE(G)=\sum^{n}_{i=1}e^{q^{}_i}$,where $q_1, q_2, \dots, q_n$are the eigenvalues of the signless Laplacian matrix of $G$. In this paper, we first characterize the unicyclic graphs with the first two largest and smallest $SLEE$ and then determine the unique unicyclic graph with maximum $SLEE$ among the unicyclic graphs on $n$ vertices with given diameter.