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For a simple graph G, the ABC-index is a degree based topological index and is defined as where dv is the degree of the vertex υ in G. Recently, the Laplacian ABC-matrix was introduced in [22] is defined by where is the diagonal matrix of ABC-degrees and Ã(G) is the ABC-matrix of G: The eigenvalues of the matrix are called the Laplacian ABC-eigenvalues of G. In the article, we consider the problem of characterization of connected graphs having exactly three distinct Laplacian ABC-eigenvalues. We solve this problem for bipartite graphs, multipartite graphs, unicyclic graphs, regular graphs and prove the non-existence of such graphs with diameter greater than 2. We introduce the concept of trace norm of the matrix called the Laplacian ABC-energy of G. We obtain some upper and lower bounds for the Laplacian ABC-energy and characterize the extremal graphs which attain these bounds. [ABSTRACT FROM AUTHOR]

The energy of a graph is defined as the sum of the absolute values of all eigenvalues with respect to its adjacency matrix. Denote by Gn,m the set of all connected graphs having n vertices and m edges. Caporossi et al. [Caporossi G, Cvetkovi D, Gutman I, et al. Variable neighbourhood search for extremal graphs. 2. Finding graphs with external energy. J Chem Inf Comput Sci. 1999;39:984–996] conjectured that among all graphs in Gn,m, n ≥ 6 and n − 1 ≤ m ≤ 2(n − 2), the graphs with minimum energy are the star Sn with m−n + 1 additional edges all connected to the same vertices for m ≤ n + ⌊ (n − 7) / 2 ⌋ , and the bipartite graph with two vertices on one side, one of which is connected to all vertices on the other side, otherwise. In this paper, we provide a new approach to investigate the conjecture above. Especially, we determine the unique tetracyclic graph having minimum energy. [ABSTRACT FROM AUTHOR]

A graphis nonsingular if its adjacency matrixis nonsingular. A nonsingular graphis said to have an inverseifis signature similar to a nonnegative matrix. An-graphis a bipartite graph with a unique perfect matching. We introduce the ‘even’ness of a nonmatching edge and show that under certain conditions a connected-graph has an inverse if and only ifis bipartite, whereis the set of all even nonmatching edges. This extends some known results, providing us with a larger class of graphs possessing inverses. [ABSTRACT FROM PUBLISHER]