Tamtam Proceedings - lamsin

A new adjacency list-matrix 591Figure 1. A directed graph G.2. Classical Graph RepresentationsAs previously stated, the standard representations are the n × n adjacency matrix, theadjacency list structure (also called edge lists and linked lists) and the incidence matrix.In this paper we examine graphs with unlabeled nodes and edges.Definition 1 A directed graph G (also called digraph or quiver) is a pair (V, E) whereV is a set of vertices and E is a set of edges where V = {v 1 , v 2 , ..., v n } ; E = {(v i , v j ) :v i , v j ∈ V } ; E ⊆ V × V ; |V | = n ; |E| = m.An undirected graph is a special case of a directed one where for any edge (v i , v j ∈ E)there exists an edge (v j , v i ∈ E). This paper deals only with directed graphs as being themost general case. We refer to Fig.1. for a geometric presentation of a graph G .Definition 2 Two vertices v i and v j are said to be adjacent, or neighbouring, if the edgejoining the two vertices belongs to E (i.e.)(v i , v j ∈ E)). Two edges are said adjacent ifthey have one common end vertex. A degree of a vertex v i is the number of edges that areincident.Definition 3 The adjacency lists are an array of vertices, each of which contains a listof all adjacent vertices (in an arbitrary order). A vertex v j appears in the i th entry of thearray if there is an edge between the vertices v i and v j . For directed graphs, we also usea successor list and a predecessor one. The successor lists (SL) store for each v i ∈ V thelist of its successors. While, the predecessor lists (PL) store for each v i ∈ V the list of itspredecessors.1 1 1 10 0 0 01 1 1 10 1 0 0Table 1. The adjacency matrix of graph G.TAMTAM –Tunis– 2005