First Zagreb Spectral Radius of Graphs with Self-Loops.

Source abstract

Let G_S be a graph with self-loops, where S is the set of all those vertices which have self-loops in G_s. Let d_i denote the degree of v_i in G_S. Then the ij-th entry of the first Zagreb matrix Z_1(G_s) is d_i'+d_j' if v_i\sim v_j or i=j and v_i\in S, and 0 otherwise. The spectral radius of Z_1(G_S) is denoted by \zeta_1(G_S). In this paper, we obtain some bounds for \zeta_1(G_S) in terms of structural and spectral parameters of G_S.