Degree based energy and spectral radius of a graph with self-loops

Abstract

Let GX be a graph obtained from a simple graph G by attaching a self-loop at each vertex of [Formula: see text]. The general extended adjacency matrix for the graph GX is defined and the bounds for the degree based energy of the graph GX are obtained. The study extends the notion of degree based energy of simple graphs to graphs with self-loops. For the graph GX of order n and size m with σ self-loops, the adjacency energy, [Formula: see text]. The spectral radius [Formula: see text] of its adjacency matrix is always less than or equal to [Formula: see text], where Δ is the maximum degree in the graph GX and the equality conditions are given for [Formula: see text]. Few more bounds for [Formula: see text] are also obtained. The study shows that, the spectral radius [Formula: see text] of its extended adjacency matrix satisfies: [Formula: see text]. We conclude the article by computing the extended adjacency spectrum of complete graph and complete bipartite graphs with self-loops.

Keywords

• Graphs with self-loop
• spectral radius
• degree based topological indices
• divisor matrix
• spectrum
• 05C50