On Spectral Radius and Energy of a Graph with Self-Loops

The spectral radius of a square matrix is the maximum among absolute values of its eigenvalues. Suppose a square matrix is nonnegative; then, by Perron-Frobenius theory, it will be one among its eigenvalues. In this paper, Perron-Frobenius theory for adjacency matrix of graph with self-loops AGS will be explored. Specifically, it discusses the nontrivial existence of Perron-Frobenius eigenvalue and eigenvector pair in the matrix AGS-σnI, where σ denotes the number of self-loops. Also, Koolen-Moulton type bound for the energy of graph GS is explored. In addition, the existence of a graph with self-loops for every odd energy is proved.