A graph $G$ of order $n$ is called $k-$step Hamiltonian for $k\geq 1$ if we can label the vertices of $G$ as $v_1,v_2,\ldots,v_n$ such that $d(v_n,v_1)=d(v_i,v_{i+1})=k$ for $i=1,2,\ldots,n-1$. The (vertex) chromatic NUMBER of a graph $G$ is the minimum NUMBER of colors needed to color the vertices of $G$ so that no pair of adjacent vertices receive the same color. The CLIQUE NUMBER of $G$ is the maximum cardinality of a set of pairwise adjacent vertices in $G$. In this paper, we study the chromatic NUMBER and the CLIQUE NUMBER in $k-$step Hamiltonian graphs for $k\geq 2$. We present upper bounds for the chromatic NUMBER in $k-$step Hamiltonian graphs and give characterizations of graphs achieving the equality of the bounds. We also present an upper bound for the CLIQUE NUMBER in $k-$step Hamiltonian graphs and characterize graphs achieving equality of the bound.

The rings considered in this article are commutative with identity which are not integral domains. Let $R$ be a ring. An ideal $I$ of $R$ is said to be an annihilating ideal of $R$ if there exists $r\in R\backslash \{0\}$ such that $Ir = (0)$. Let $\mathbb{A}(R)$ denote the set of all annihilating ideals of $R$ and let $\mathbb{A}(R)^{*} = \mathbb{A}(R)\backslash \{(0)\}$. Recall that the annihilating-ideal graph of $R$, denoted by $\mathbb{AG}(R)$, is an undirected graph whose vertex set is $\mathbb{A}(R)^{*}$ and distinct vertices $I$ and $J$ are adjacent in this graph if and only if $IJ = (0)$. The aim of this article is to characterize zero-dimensional rings such that the CLIQUE NUMBER of their annihilating-ideal graphs is at most four.

The annihilator graph AG (R) of a commutative ring R is a simple undirected graph with the vertex setZ (R)* and two distinct vertices are adjacent if and only if ann (x) È ann (y)¹ann (xy). In this paper we give the sufficient condition for a graph AG (R) to be complete. We characterize rings for which AG (R) is a regular graph, we show that g (AG (R)) Î {1, 2} g and we also characterize the rings for which AG (R) has a cut vertex. Finally we find the CLIQUE NUMBER of a finite reduced ring and characterize the rings for which AG (R) is a planar graph.

The CLIQUE polynomial of a graph G is the ordinary generating function of the NUMBER of complete subgraphs (CLIQUEs) of G. In this paper, we introduce a new vertex-weighted version of these polynomials. We also show that these weighted CLIQUE polynomials have always a real root provided that the weights are non-negative real NUMBERs. As an application, we obtain a no-homomorphism criteria based on the largest real root of our vertex-weighted CLIQUE polynomial.