We study the weight distribution of a perfect coloring (equitable partition) of a graph with respect to a completely regular code (in particular, with respect to a vertex if the graph is distance-regular). We show how to compute this distribution by the knowledge of the percentage of the colors over the code. For some partial cases of completely regular codes we derive explicit formulas of weig...

Let G1 be a planar graph such that all cycles of length at most 4 are independent and let G2 be a planar graph without 3-cycles and adjacent 4-cycles. It is proved that the set of vertices of G1 and G2 can be equitably partitioned into t subsets for every t ≥ 3 so that each subset induces a forest. These results partially confirm a conjecture of Wu, Zhang

Prime graph of a ring R is a graph whose vertex set is the whole set R any any two elements $x$ and $y$ of $R$ are adjacent in the graph if and only if $xRy = 0$ or $yRx = 0$.  Prime graph of a ring is denoted by $PG(R)$. Directed prime graphs for non-commutative rings and connectivity in the graph are studied in the present paper. The diameter and girth of this graph are also studied in the pa...

We study the weight distribution of a perfect coloring (equitable partition) of a graph with respect to a completely regular code (in particular, with respect to a vertex if the graph is distance-regular). We show how to compute this distribution by the knowledge of the percentage of the colors over the code. For some partial cases of completely regular codes we derive explicit formulas of weig...

Let G = (V,E) be a connected graph, An equitable dominating S of a graph G is called the neighborhood connected equitable dominating set (nced-set) if the induced subgraph 〈Ne(S)〉 is connected The minimum cardinality of a nced-set of G is called the neighborhood connected equitable domination number of G and is denoted by γnce(G). In this paper we initiate a study of this parameter. For any gra...

Let $G=(V(G),E(G))$ be a simple, finite and undirected graph of order $n$. A $k$-vertex weightings of a graph $G$ is a mapping $w: V(G) to {1, ldots, k}$. A $k$-vertex weighting induces an edge labeling $f_w: E(G) to N$ such that $f_w(uv)=w(u)+w(v)$. Such a labeling is called an {it edge-coloring k-vertex weightings} if $f_{w}(e)not= f_{w}(echr(chr(chr('39')39chr('39'))39chr(chr('39')39chr('39'...

Coloring graphs is one of important and frequently used topics in diverse sciences. In the majority of the articles, it is intended to find a proper bound for vertex coloring, edge coloring or total coloring in the graph. Although it is important to find a proper algorithm for graph coloring, it is hard and time-consuming too. In this paper, a new algorithm for vertex coloring, edge coloring an...