For n ∈ N and D ⊆ N, the distance graph P n has vertex set {0, 1, . . . , n− 1} and edge set {ij | 0 ≤ i, j ≤ n − 1, |j − i| ∈ D}. Note that the important and very well-studied circulant graphs coincide with the regular distance graphs. A fundamental result concerning circulant graphs is that for these graphs, a simple greatest common divisor condition, their connectivity, and the existence of ...

We review some methods of associating graphs to rings. The main emphasis is on zero-divisor graphs and comaximal ones. Some new results in both directions are presented. AMS Mathematics Subject Classification (2000): 13A99, 05C99, 16S50

Let S be a finite, nonempty set of positive integers. Then, the divisor graph ) (S D of S has S as its vertex set, and vertices i and j are adjacent if and only if either j i or i j . This paper investigates the vertex-chromatic number, the clique number, the clique cover number, and the independence number of ]) ([n D and its complement, where } , 1 : { ] [ Ν ∈ ≤ ≤ = n n i i n . Besides, we di...

This paper establishes a set of theorems that describe the diameter of a zero-divisor graph for a finite direct product R1 × R2 × · · · × Rn with respect to the diameters of the zero-divisor graphs of R1, R2, · · · , Rn−1 and Rn(n > 2).

A (finite or infinite) complete bipartite graph together with some end vertices all adjacent to a common vertex is called a complete bipartite graph with a horn. For any bipartite graph G, we show that G is the graph of a commutative semigroup with 0 if and only if it is one of the following graphs: star graph, two-star graph, complete bipartite graph, complete bipartite graph with a horn. We a...