2 research outputs found
Spectra of the subdivision-vertex and subdivision-edge coronae
The subdivision graph of a graph is the graph obtained
by inserting a new vertex into every edge of . Let and be two
vertex disjoint graphs. The \emph{subdivision-vertex corona} of and
, denoted by , is the graph obtained from
and copies of , all vertex-disjoint, by joining the th
vertex of to every vertex in the th copy of . The
\emph{subdivision-edge corona} of and , denoted by , is the graph obtained from and copies of
, all vertex-disjoint, by joining the th vertex of to every
vertex in the th copy of , where is the set of inserted
vertices of . In this paper we determine the adjacency
spectra, the Laplacian spectra and the signless Laplacian spectra of (respectively, ) in terms of the corresponding
spectra of and . As applications, the results on the spectra of
(respectively, ) enable us to construct
infinitely many pairs of cospectral graphs. The adjacency spectra of (respectively, ) help us to construct many infinite
families of integral graphs. By using the Laplacian spectra, we also obtain the
number of spanning trees and Kirchhoff index of and
, respectively.Comment: 15 pages, 3 figure. arXiv admin note: text overlap with
arXiv:1212.0851, arXiv:1212.0619; and with 1209.5906 by other author
On the -spectra of some join graphs
Let be a simple, connected graph and let be the adjacency matrix
of . If is the diagonal matrix of the vertex degrees of , then for
every real , the matrix is defined as
The eigenvalues of the
matrix form the -spectrum of . Let , , and denote the subdivision-vertex
join, subdivision-edge join, -vertex join and -edge join of two graphs
and , respectively. In this paper, we compute the
-spectra of , , and for a
regular graph and an arbitrary graph in terms of their
-eigenvalues. As an application of these results, we construct
infinitely many pairs of -cospectral graphs