On the eigenvalues of Laplacian ABC-matrix of graphs

For a simple graph G, the ABC-index is a degree based topological index and is defined aswhere 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.