Abstract—The aim of the paper is to define the adjacency matrix of a Semigraph. We expect to see many applications of this in the future. Incidence matrix of a semigraph has been defined by the authors[4].Here we not only define Adjacency Matrix associated with a semigraph, but also state necessary and sufficient conditions for a matrix to be semigraphical. At the end of the paper spectrum of a semigraph is defined and some of its spectral properties are studied. Index Terms—Adjacency matrix of a semigraph, spectrum of a semigraph. I. INTRODUCTION Semigraphs were introduced by E.Sampathkumar[2] as generalization of graphs. Many authors [2], [3], [4] have studied properties of semigraphs. Semigraphs prove to be a better model than graphs in all those applications where instead of two points to be connected by an edge we need to connect several points by an edge and the order in which they appear in the edge is of prime importance. Unique representation of any discrete structure in matrix form is important for applications in computer science. In Section 2 we give some preliminary definitions and in Section 3 we give main definition and necessary and sufficient conditions for a matrix to be adjacency matrix of a semigraph. Some spectral properties of adjacency matrix are discussed in Section 4. II. DEFINITIONS Semigraph: A semigraph G is a pair (V, X) where V is a non-empty set whose elements are called vertices of G and X is a set of n- tuples, called edges of G, of distinct vertices, for various n (n ≥ 2) satisfying the following conditions: (i) Any two edges have at most one vertex in common (ii) Two edges (u 1 , u 2 , …, u n ) and (v 1 , v 2 , v m ) are considered to be equal if (a) m = n and (b) either u i = v i for i = 1, 2, …, n or u i = v n+1-i , for i = 1, 2, …, n For the edge E = (u 1 , u 2 , …u n ), u 1 and u n are called the end vertices of E and u 2 , u 3 , …u n-1 are called the middle vertices of E. Adjacent vertices: Manuscript received April 14, 2012; revised May 26, 2012. C. M. Deshpande is with the College of Engineering, Pune (e-mail: hod.maths@coep.ac.in). Y. S. Gaidhani is with the M. E. S. Abasaheb Garware College, Pune. (e-mail: yogeshrigaidhani@gmail.com). Two vertices in a semigraph are said to be adjacent if they belong to the same edge and are said to be consecutively adjacent if in addition they are consecutive in order as well. For example, in the semigraph G of Fig.1, the vertices v 1 and v 4 are adjacent while the vertices v 2 and v 3 are consecutively adjacent. Cardinality: Cardinality of an edge E in a semigraph G is the number of vertices lying on that edge. In the semigraph G of Fig.1, cardinality of E 1 , denoted as |E 1 |, is 4. Adjacent Edges: Two edges E 1 and E 2 in a semigraph G are said to be adjacent if they have a vertex in common. For instance, in the semigraph G of Fig.1, the edges E 2 and E 3 are adjacent. p-edge: p-edge (partial edge) of E is a (k-j+1)-tuple E'' = (v ij , v ij+1 ,…, v ik ) where 1 ≤ j < k ≤ n. f-edge: f-edge is any edge of a semigraph, called full edge. fp-edge is an edge which is either an f-edge or a p-edge. A semigraph G may be drawn as a set of points representing the vertices. An edge E = (v i1 ,v i2 ,…,v ir ) is represented by a Jordan curve joining the points corresponding to the vertices v i1 ,v i2 ,…,v ir in the same order as they appear in E. The end points of the curve (i.e. the end vertices of E) are denoted by thick dots. The points lying on an edge in between the end points (i.e. middle vertices of E) are denoted by small hollow circles. If an end vertex v of an edge E is a middle vertex of some edge E', a small tangent is drawn to the circle (representing v on E') at the end of E. Fig. 1 represents the semigraph G. E 1 = (v 1 , v 2 , v 3 , v 4 ), E 2 = (v 1 , v 6 , v 5 ), E 3 = (v 4 , v 6 , v 7 ), E 4 = (v 4 , v 5 ), E 5 = (v 5 , v 9 ), E 6 = (v 5 , v 7 ) Fig. 1. Semigraph G. A. Remarks:  A semigraph with n = 2 is a graph. Hence it can be seen as a natural generalization of graphs.  A semigraph is a linear hypergraph H with an order given to each edge of H. The adjacency matrix associated with a semigraph as defined in this paper can shed more light on the properties of linear hypergraphs. About Adjacency Matrix of Semigraphs C. M. Deshpande and Y. S. Gaidhani International Journal of Applied Physics and Mathematics, Vol. 2, No. 4, July 2012 250