On the Line Graph for Zero-Divisors of

Let be a completely regular Hausdorff space and let be the ring of all continuous real valued functions defined on . In this paper, the line graph for the zero-divisor graph of is studied. It is shown that this graph is connected with diameter less than or equal to 3 and girth 3. It is shown that this graph is always triangulated and hypertriangulated. It is characterized when the graph is complemented. It is proved that the radius of this graph is 2 if and only if has isolated points; otherwise, the radius is 3. Bounds for the dominating number and clique number are also found in terms of the density number of . 1. Introduction Let be a completely regular Hausdorff space and the ring of all continuous real valued functions defined on . For each , let , , , and . For all notations and undefined terms concerning the ring , the reader may consult [1]. If , then is a field isomorphic to . So we will assume that . Let be a commutative ring. is the set of zero-divisors of , and . The zero-divisor graph of , , usually written as , is the graph in which each element of is a vertex, and two distinct vertices and are adjacent if and only if . For further details about this graph, see [2] and the survey [3] for a list of references. The line graph of a graph , denoted by , is a graph whose vertices are the edges of and two vertices of are adjacent wherever the corresponding edges of are incident to a common vertex; see [4]. In this case, if are adjacent vertices in , then is a vertex in . For any undefined terms in graph theory, the reader may consult [5]. The zero-divisor graph for was introduced and studied in [6]. A more general study for reduced rings was done in [7]. In this paper we will study the line graph for the zero-divisor graph . An element if and only if . Let . Then is a vertex in if . Since is an undirected graph, then . We will study when is connected and calculate its diameter and girth. We will show that is always triangulated and hypertriangulated and characterize when is complemented. We will find the radius and give bounds for the dominating and clique numbers. 2. Connectedness Let be a graph and let and be two distinct vertices in . The distance？？ between and is the length of the shortest path joining them in ; if no such path exists, we set . The associate number？？ of a vertex of a graph is defined to be . A vertex is center in if for any vertex . The radius of is defined to be and the diameter of is . The graph is connected if any two of its vertices are linked by a path in ; otherwise is disconnected. In this section, we will show that