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...

The detour pebbling number of a graph \(G\) is the least positive integer \(f^{*}(G)\) such that these pebbles are placed on vertices \(G\), we can move pebble to target vertex by sequence moves each taking two off and placing one an adjacent using path. In this paper, compute for commutative ring zero-divisor graphs, sum product zero divisor graphs.

Let R be a commutative ring and let ? Z n the zero divisor graph of id="M2"> R , whose vertices are nonzero divisors id="M3"> such that two id="M4"> u , v adjacen...

Let R be a commutative ring with nonzero identity and Z(R) its set of zero-divisors. The new extension the zero-divisor graph $$\widetilde{\Gamma }(R)$$ vertices $$Z(R)^{\star }$$ , two distinct x y are adjacent if only $$xy=0$$ or $$x+y\in Z(R)$$ . In this article, we study, in general case, For any R, provide sufficient necessary conditions for }(R[x_1,\dots x_n])$$ to complete. At last, pres...

Let R be a commutative ring with non-zero identity. The cozero-divisor graph of R, denoted by Γ′(R), is a graph with vertices in W ∗(R), which is the set of all non-zero and non-unit elements of R, and two distinct vertices a and b in W ∗(R) are adjacent if and only if a / ∈ bR and b / ∈ aR. In this paper, we characterize all commutative rings whose cozero-divisor graphs are forest, star, doubl...

In this paper, we introduce the multiplicative zero-divisor graph of a multiplicative lattice and study Beck-like coloring of such graphs. Further, it is proved that for such graphs, the chromatic number and the clique number need not be equal. On the other hand, if a multiplicative lattice L is reduced, then the chromatic number and the clique number of the multiplicative zero-divisor graph of...