In an earlier paper, the present authors (2013) [1] introduced the alternating chromatic number for hypergraphs and used Tucker’s Lemma, an equivalent combinatorial version of the Borsuk-Ulam Theorem, to show that the alternating chromatic number is a lower bound for the chromatic number. In this paper, we determine the chromatic number of some families of graphs by specifying their alternating...

The Narumi-Katayama index of a graph G is equal to the product of degrees of all the vertices of G. In this paper, we examine the NarumiKatayama index of some derived graphs such as a Mycielski graph, subdivision graphs, double graph, extended double cover graph, thorn graph, subdivision vertex join and edge join graphs.

Let n(G) denote the number of vertices of a graph G and let (G) be the independence number of G, the maximum number of pairwise nonadjacent vertices of G. The Hall ratio of a graph G is defined by (G)=max { n(H) (H) : H ⊆ G } , where the maximum is taken over all induced subgraphs H of G. It is obvious that every graph G satisfies (G) (G) (G) where and denote the clique number and the chromatic...

let $r$ be a commutative ring with identity. an ideal $i$ of a ring $r$is called an annihilating ideal if there exists $rin rsetminus {0}$ such that $ir=(0)$ and an ideal $i$ of$r$ is called an essential ideal if $i$ has non-zero intersectionwith every other non-zero ideal of $r$. thesum-annihilating essential ideal graph of $r$, denoted by $mathcal{ae}_r$, isa graph whose vertex set is the set...

in this paper, it was shown that , where  and , and , where  is not prime and , are od-characterizable.

Let G be a group. Recall that the intersection graph of subgroups of G is an undirected graph whose vertex set is the set of all nontrivial subgroups of G and distinct vertices H,K are joined by an edge in this graph if and only if the intersection of H and K is nontrivial. The aim of this article is to investigate the interplay between the group-theoretic properties of a finite group G and the...

A set $S$ of vertices is a determining for graph $G$ if every automorphism uniquely determined by its action on $S$. The size smallest called number, $Det(G)$. said to be $d$-distinguishable there coloring the with $d$ colors so that only trivial preserves color classes. such distinguishing $Dist(G)$. If $Dist(G) = 2$, cost 2-distinguishing, $\rho(G)$, class over all 2-distinguishing colorings ...