Let P = {V 1 , V 2 , V 3 , . . . , V k } be a partition of vertex set V (G) of order k ≥ 2. For all V i and V j in P , i = j, remove the edges between V i and V j in graph G and add the edges between V i and V j which are not in G. The graph G P k thus obtained is called the k−complement of graph G with respect to a partition P . For each set V r in P , remove the edges of graph G inside V r and add the edges of G (the complement of G) joining the vertices of V r . The graph G P k(i) thus obtained is called the k(i)−complement of graph G with respect to a partition P . In this paper, we study Laplacian energy of generalized complements of some families of graph. An effort is made to throw some light on showing variation in Laplacian energy due to changes in a partition of the graph.

This study aims to extend the notion of degree-based topological index, associated adjacency-type matrix and its energy from a simple graph to a graph with self-loops. Let GS be a graph with k self-loops obtained from a simple graph G, we define Sombor index for GS as SO(GS)where S ⊆ V (G) having self-loop to each of its vertices in S. In addition we investigate some fundamental properties of Sombor eigenvalues, McClelland and Koolen-Moulton-type bound for Sombor energy of GS. Also explores the correlation between Sombor energy of GS and the total π−electron energies associated with the corresponding hetero-molecular systems.

Let $G$ be a simple connected graph. The energy of a graph $G$ is defined as sum of the absolute eigenvalues of an adjacency matrix of the graph $G$. It represents a proper generalization of a formula valid for the total $\pi$-electron energy of a conjugated hydrocarbon as calculated by the Huckel molecular orbital (HMO) method in quantum chemistry. A coloring of a graph $G$ is a coloring of its vertices such that no two adjacent vertices share the same color. The minimum number of colors needed for the coloring of a graph $G$ is called the chromatic number of $G$ and is denoted by $\chi(G)$. The color energy of a graph $G$ is defined as the sum of absolute values of the color eigenvalues of $G$. The graphs with large number of edges are referred as cluster graphs. Cluster graphs are graphs obtained from complete graphs by deleting few edges according to some criteria. It can be obtained on deleting some edges incident on a vertex, deletion of independent edges/triangles/cliques/path P3 etc. Bipartite cluster graphs are obtained by deleting few edges from complete bipartite graphs according to some rule. In this paper, the color energy of cluster graphs and bipartite cluster graphs are studied.

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