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

Let GX be a graph obtained from a simple graph G by attaching a self-loop at each vertex of X⊆V(G). 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, E(GX)≥4mn+2σ(n−σ)n. The spectral radius ρ(GX) of its adjacency matrix is always less than or equal to 12−1+1+4(2m+σ+Δ−1), where Δ is the maximum degree in the graph GX and the equality conditions are given for 0<σ<n. Few more bounds for ρ(GX) are also obtained. The study shows that, the spectral radius fmax(GX) of its extended adjacency matrix satisfies: mini∼jFij(2m+σ)n≤fmax(GX)≤maxi∼jFij2(2m+σ). We conclude the article by computing the extended adjacency spectrum of complete graph and complete bipartite graphs with self-loops.