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 asABC(G) = sigma(vivj is an element of E(G)) root d(vi) + d(vj) - 2/d(vi)d(vj),where d(v )is the degree of the vertex v in G. Recently, the Laplacian ABC-matrix was introduced in [22] is defined by (sic)L(G) = (sic)D(G) - (sic)A(G), where D(G) is the diagonal matrix of ABC-degrees and (sic)A(G) is the ABC-matrix of G. The eigenvalues of the matrix (sic)L(G) 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 (sic)L(G) - tr((sic)LG))/nI, 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.