On the Estrada index of unicyclic and bicyclic signed graphs

TL;DR

This paper investigates the Estrada index of signed graphs, characterizing those with maximum values among unicyclic and bicyclic graphs, especially focusing on unbalanced graphs with the pairing property and complete bipartite graphs.

Contribution

It provides a characterization of signed unicyclic graphs with maximum Estrada index and identifies extremal graphs within specific classes of unbalanced signed graphs.

Findings

Maximum Estrada index graphs among unbalanced unicyclic graphs identified.

Signed graphs with the pairing property and maximum Estrada index characterized.

Extremal signed complete bipartite graphs determined.

Abstract

Let $\Gamma=(G, \sigma)$ be a signed graph of order $n$ with eigenvalues $\mu_1,\mu_2,\ldots,\mu_n.$ We define the Estrada index of a signed graph $\Gamma$ as $EE(\Gamma)=\sum_{i=1}^ne^{\mu_i}$. We characterize the signed unicyclic graphs with the maximum Estrada index. The signed graph $\Gamma$ is said to have the pairing property if $\mu$ is an eigenvalue whenever $-\mu$ is an eigenvalue of $\Gamma$ and both $\mu$ and $-\mu$ have the same multiplicities. If $\Gamma_{p}^-(n, m)$ denotes the set of all unbalanced graphs on $n$ vertices and $m$ edges with the pairing property, we determine the signed graphs having the maximum Estrada index in $\Gamma_{p}^-(n, m)$, when $m=n$ and $m=n+1$. Finally, we find the signed graphs among all unbalanced complete bipartite signed graphs having the maximum Estrada index.