## Spectra of a New Join in Duplication Graph

**Authors:** K. Reji Kumar, Renny P. Varghese

The Duplication graph DG of a graph G, is obtained by inserting
new vertices corresponding to each vertex of G and making the vertex adja-
cent to the neighbourhood of the corresponding vertex of G and deleting the
edges of G. Let G1 and G2 be two graph with vertex sets V (G1) and V (G2)
respectively. The DG - vertex join of G1 and G2 is denoted by G1 t G2 and
it is the graph obtained from DG1 and G2 by joining every vertex of V (G1)
to every vertex of V (G2). The DG - add vertex join of G1 and G2 is denoted
by G1 ./ G2 and is the graph obtained from DG1 and G2 by joining every
additional vertex of DG1 to every vertex of V (G2). In this paper we determine
the A - spectra and L - spectra of the two new joins of graphs for a regular
graph G1 and an arbitrary graph G2 . As an application we give the number
of spanning tree, the Kirchhoff index and Laplace energy like invariant of the
new join. Also we obtain some infinite family of new class of integral graphs

**Comments:** 10 Pages.

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### Submission history

[v1] 2016-10-04 22:15:42

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