e-ADJACENCY MATRIX AND e-LAPLACIAN MATRIX OF SEMIGRAPH

In this paper we define e-Adjacency matrix Ae(S) and e-Laplacian matrix of a Semigraph Le(S). Also discuss some results of eigenvalues of these matrices. We define e-Energy of Semigraph Ee(S) using eigenvalues of its e-adjacency matrix and e-Laplacian energy of Semigraph LEe(S) using eigenvalues of its e-Laplacian matrix. We investigate relation between e-energyEe(S) and e-Laplacian energy LEe(S) for regular Semigraphs.


INTRODUCTION
The concept of Semigraph was first introduced by E Sampathkumar [1] in 2000 as a generalization of a graph where an edge can have more than two vertices. In last two decades lot of research has been done in this area of Semigraph. Many concepts of graphs have been generalized to Semigraph. Domination in Semigraph has been extensively studied by S S Kamath, R S Bhat, V.

Cardinality of edge:
It is the number of vertices in an edge and it is denoted as| |.
Adjacent vertices: Two vertices are said to be adjacent if they belong to same edge and they are said to be consecutively adjacent if they belong to same edge and are consecutive in order.
Adjacent edges: Two edges are said to be adjacent if there is a vertex in common.

Special Types of Semigraphs [2]
Regular Semigraph: A semigraph S(V,E) is said to be regular if all its vertices have the same degree of a particular type of degree.
r-uniform Semigraph: A semigraph S(V,E) is said to be r-uniform if cardinality of each edge in S is r.
Complete Semigraph: If any two vertices in a semigraph S(V,E) are adjacent then the semigraph S(V,E) is said to be complete.
Following are some of the families of semigraph which are complete.
1) a semigraph consisting of a single s-edge of cardinality r.
2)) −1 1 a semigraph consisting of an s-edge of cardinality r-1 and one vertex joined with each of the r-1 vertices by an edge of cardinality two.
Incidence matrix and Adjacency matrix of a semigraph has been studied byY S Gaidhani and C M Deshpande see [3], [4].

MAIN RESULTS
In first section of this paper we define e-Adjacent degree of a vertex ( ( )). Also we define an e-Adjacency matrix, e-Degree matrix and e-Laplacian matrix of a Semigraph and discuss some results of eigenvalues of these matrices. In the second section of this paper we define e-energy of Semigraph using eigenvalues of its e-adjacency matrix and e-Laplacian energy of Semigraph using eigenvalues of its e-Laplacian matrix.  We observe that Adjacent degree of a vertex ( ) is same as e-Adjacent degree of a vertex

Example. 8.5
Let be a semigraph with n vertices consisting of a single s-edge of cardinality n.
where u is end vertex or middle end vertex of semigraph S(V,E).

Matrix Energy and Polynomial energy of a Semigraph has been defined by Y S Gaidhani and CM
Deshpande considering adjacency matrix of Semigraph see [6].
Here we define e-Energy of Semigraph with respect to e-Adjacency matrix. Also we define e-Laplacian energy of Semigraph S(V,E).