Some Properties on Estrada Index of Folded Hypercubes Networks

Abstract

Let $G$ be a simple graph with $n$ vertices and let ${\lambda }_1,{\lambda }_2,\dots ,{\lambda }_n$ be the eigenvalues of its adjacency matrix; the Estrada index $EE\left(G\right)$ of the graph $G$ is defined as the sum of the terms $e^{\lambda i}$, $i=\mathrm{1,2},\dots ,n$. The $n$-dimensional folded hypercube networks $F{Q}_n$ are an important and attractive variant of the $n$-dimensional hypercube networks $Q_n$, which are obtained from $Q_n$ by adding an edge between any pair of vertices complementary edges. In this paper, we establish the explicit formulae for calculating the Estrada index of the folded hypercubes networks $F{Q}_n$ by deducing the characteristic polynomial of the adjacency matrix in spectral graph theory. Moreover, some lower and upper bounds for the Estrada index of the folded hypercubes networks $F{Q}_n$ are proposed.