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

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