On the $c$-dominating Estrada index of a graph

Abstract

Let $G$ be a graph with $n$ vertices, then the $c$-dominating matrix of $G$ is the square matrix of order $n$ whose $(i,j)$-entry is equal to 1 if the $i$-th and $j$-th vertex of $G$ are adjacent, is also equal to 1 if $i=j$, $v_i\in D_c$, and zero otherwise.

The $c$-dominating energy of a graph $G$, is defined as the sum of the absolute values of all eigenvalues of the $c$-dominating matrix.

The main purposes of this paper are to introduce the $c$-dominating Estrada index of a graph. Moreover, to obtain upper and lower bounds for the $c$-dominating Estrada index and investigate the relations between the $c$-dominating Estrada in