Signless Laplacian Energy of a Graph with Self-Loops

Let G_S be a graph obtained by adding a self-loop to each vertex of S ⊆ V in a graph G(V, E). The signless Laplacian matrix of a graph G_S containing |S|=σ self-loops is, Q(G_S)=A(G_S) + D(G), where A(G_S), D(G) are the adjacency matrix of G_S and diagonal matrix of G respectively. The signless Laplacian energy of a graph G_S(n, m) containing σ self-loops and having q_i, i=1, 2, …, n, as signless Laplacian eigenvalues is defined as (Formula Presented). In this n paper, the signless Laplacian matrix of a graph with self-loops is considered. Some basic spectral properties and bounds for signless Laplacian energy are studied. The signless Laplacian spectral properties of complete graph, complete bipartite graph, and star graph with self-loops are also obtained. Correlation between Signless Laplacian energy and total π−electron energy of hetero-molecules is obtained.