33 research outputs found
Signless Laplacian determinations of some graphs with independent edges
{Signless Laplacian determinations of some graphs with independent edges}%
{Let be a simple undirected graph. Then the signless Laplacian matrix of
is defined as in which and denote the degree matrix
and the adjacency matrix of , respectively. The graph is said to be
determined by its signless Laplacian spectrum ({\rm DQS}, for short), if any
graph having the same signless Laplacian spectrum as is isomorphic to .
We show that is determined by its signless Laplacian spectra
under certain conditions, where and denote a natural number and the
complete graph on two vertices, respectively. Applying these results, some {\rm
DQS} graphs with independent edges are obtained
Distance matrices on the H-join of graphs: A general result and applications
Given a graph with vertices and a set of pairwise vertex disjoint graphs the vertex of is assigned to Let be the graph obtained from the graphs and the edges connecting each vertex of with all the vertices of for all edge of The graph is called the of . Let be a matrix on a graph . A general result on the eigenvalues of , when the all ones vector is an eigenvector of for , is given. This result is applied to obtain the distance eigenvalues, the distance Laplacian eigenvalues and as well as the distance signless Laplacian eigenvalues of when are regular graphs. Finally, we introduce the notions of the distance incidence energy and distance Laplacian-energy like of a graph and we derive sharp lower bounds on these two distance energies among all the connected graphs of prescribed order in terms of the vertex connectivity. The graphs for which those bounds are attained are characterized.publishe