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

## Main Article Content

## 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.