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.

In this paper, we first extend the weighted handshaking lemma, using a generalization of the concept of the degree of vertices to the values of graphs. This edge-version of the weighted handshaking lemma yields an immediate generalization of the Mantel's classical result which asks for the maximum number of edges in triangle-free graphs to the class of K4-free graphs. Then, by defining the concept of value for cliques (complete subgraphs) of higher orders, we also extend the classical result of Mantel for any graph G. We finally conclude our paper with a discussion about the possible future works.