# Combinatorics and Graph Theory

## 1610 Submissions

 viXra:1610.0050 [pdf] submitted on 2016-10-04 21:54:55

### Spectra of New Join of Two Graphs

Authors: Renny P Varghese, K Rejikumar
Comments: 8 Pages.

Let G1 and G2 be two graph with vertex sets V (G1); V (G2) and edge sets E(G1);E(G2) respectively. The subdivision graph S(G) of a graph G is the graph obtained by inserting a new vertex into every edges of G. The SGvertexjoin of G1 and G2 is denoted by G1}G2 and is the graph obtained from S(G1) [ G1 and G2 by joining every vertex of V (G1) to every vertex of V (G2). In this paper we determine the adjacency spectra ( respectively Laplacian spectra and signless Laplacian spectra) of G1}G2 for a regular graph G1 and an arbitrary graph G2
Category: Combinatorics and Graph Theory

 viXra:1610.0049 [pdf] submitted on 2016-10-04 22:15:42

### Spectra of a New Join in Duplication Graph

Authors: K. Reji Kumar, Renny P. Varghese
Comments: 10 Pages.

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
Category: Combinatorics and Graph Theory

 viXra:1610.0043 [pdf] submitted on 2016-10-04 11:30:07

### Spectrum of (K; r) Regular Hypergraph

Authors: K Reji Kumar, Renny P Varghese
Comments: 8 Pages.

We present a spectral theory of uniform, regular and linear hyper- graph. The main result are the nature of the eigen values of (k; r) - regular linear hypergraph and the relation between its dual and line graph. We also discuss some properties of Laplacian spectrum of a (k; r) - regular hypergraphs.
Category: Combinatorics and Graph Theory