On the \(c\)-dominating Estrada index of a graph

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