# Combinatorics and Graph Theory

## 1610 Submissions

[3] **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

[2] **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

[1] **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