On the average degree eigenvalues and average degree energy of graphs

Abstract

Given a graph $G$ with $n$ vertices $v_1, v_2, \dots, v_n$ and the vertex degrees $d_1, d_2,...,d_n$ respectively. We associate to $G$ an average degree matrix $A_v(G)$ whose $(i,j)^{th}$ entry is $\frac{d_i + d_j}{2}$. We explore some properties of the eigenvalues and energy of $A_v(G)$.