Effects on the distance Laplacian spectrum of graphs with clusters by adding edges

Abstract

All graphs considered are simple and undirected. A cluster in a graph is a pair of vertex subsets (C, S), where C is a maximal set of cardinality |C| ≥ 2 of independent vertices sharing the same set S of |S| neighbors. Let G be a connected graph on n vertices with a cluster (C, S) and H be a graph of order |C|. Let G(H) be the connected graph obtained from G and H when the edges of H are added to the edges of G by identifying the vertices of H with the vertices in C. It is proved that G and G(H) have in common n −|C| + 1 distance Laplacian eigenvalues, and the matrix having these common eigenvalues is given, if H is the complete graph on |C| vertices then ∂ −|C| + 2 is a distance Laplacian eigenvalue of G(H) with multiplicity|C| − 1, where ∂ is the transmission in G of the vertices in C. Furthermore, it is shown that if G is a graph of diameter at least 3, then the distance Laplacian spectral radii of G and G(H) are equal, and if G is a graph of diameter 2, then conditions for the equality of these spectral radii are established. Finally, the results are extended to graphs with two or more disjoint clusters.