On the Estrada index of unicyclic and bicyclic signed graphs

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.