For a finite bounded lattice £, we associate a zero-divisor graph G(£) which is a natural generalization of the concept of zero-divisor graph for a Boolean algebra. Also, we study the interplay of lattice-theoretic properties of £ with graph-theoretic properties of G(£). (2010). Primary 05C75, 06E99; Secondary 13M99.
Mathematics Subject Classification
The distinguishing number D(G) of a graph G is the minimum number of colors needed to color the vertices of G such that the coloring is preserved only by the trivial automorphism. In this paper we improve results about the distinguishing number of Cartesian products of finite and infinite graphs by removing restrictions to prime or relatively prime factors.
Let G be a connected graph of order n having ε(G) end-vertices. Given a positive integer t, we denote by S(G, t) the t-th generalized Sierpiński graph of G. In this note we show that if every internal vertex of G is a cut vertex, then the strong metric dimension of S(G, t) is given by dim s (S(G, t)) = ε(G) n t − 2n t−1 + 1 − n + 1 n − 1 .
Let R be a ring (not necessary commutative) with non-zero identity. The unit graph of R, denoted by G(R), is a graph with elements of R as its vertices and two distinct vertices a and b are adjacent if and only if a + b is a unit element of R. It was proved that if R is a commutative ring and m is a maximal ideal of R such that |R/m| = 2, then G(R) is a complete bipartite graph if and only if (R, m) is a local ring. In this paper we generalize this result by showing that if R is a ring (not necessary commutative), then G(R) is a complete r-partite graph if and only if (R, m) is a local ring and r = |R/m| = 2 n , for some n ∈ N or R is a finite field. Among other results we show that if R is a left Artinian ring, 2 ∈ U (R) and the clique number of G(R) is finite, then R is a finite ring.2000 Mathematics Subject Classification. 05C25, 13E10.
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