On the local irregularity vertex coloring of volcano, broom, parachute, double broom and complete multipartite graphs

2021 ◽  
Author(s):  
Arika Indah Kristiana ◽  
Nafidatun Nikmah ◽  
Dafik ◽  
Ridho Alfarisi ◽  
M. Ali Hasan ◽  
...  
Keyword(s):  
Chromatic Number ◽  
Connected Graph ◽  
Vertex Coloring ◽  
Minimum Cardinality ◽  
Label Function ◽  
Multipartite Graphs ◽  
Complete Multipartite Graphs ◽  
Vertex Set ◽  
Complete Multipartite ◽  
Local Irregularity

Let [Formula: see text] be a simple, finite, undirected, and connected graph with vertex set [Formula: see text] and edge set [Formula: see text]. A bijection [Formula: see text] is label function [Formula: see text] if [Formula: see text] and for any two adjacent vertices [Formula: see text] and [Formula: see text], [Formula: see text] where [Formula: see text] and [Formula: see text] is set ofvertices adjacent to [Formula: see text]. [Formula: see text] is called local irregularity vertex coloring. The minimum cardinality of local irregularity vertex coloring of [Formula: see text] is called chromatic number local irregular denoted by [Formula: see text]. In this paper, we verify the exact values of volcano, broom, parachute, double broom and complete multipartite graphs.

scholarly journals Sharply Transitive $1$-Factorizations of Complete Multipartite Graphs

10.37236/349 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Giuseppe Mazzuoccolo ◽  
Gloria Rinaldi
Keyword(s):  
Complete Multipartite Graph ◽  
Transitive Action ◽  
Multipartite Graphs ◽  
Complete Multipartite Graphs ◽  
Vertex Set ◽  
Even Order ◽  
Multipartite Graph ◽  
A Finite Group ◽  
Complete Multipartite ◽  
Sharply Transitive

Given a finite group $G$ of even order, which graphs $\Gamma$ have a $1$-factorization admitting $G$ as automorphism group with a sharply transitive action on the vertex-set? Starting from this question, we prove some general results and develop an exhaustive analysis when $\Gamma$ is a complete multipartite graph and $G$ is cyclic.

scholarly journals On groups admitting no integral Cayley graphs besides complete multipartite graphs

2013 ◽  
Vol 7 (1) ◽  
pp. 119-128 ◽  
Author(s):  
Alireza Abdollahi ◽  
Mojtaba Jazaeri
Keyword(s):  
Cayley Graph ◽  
Identity Element ◽  
Simple Groups ◽  
Complete Multipartite Graph ◽  
Arbitrary Graph ◽  
Multipartite Graphs ◽  
Complete Multipartite Graphs ◽  
Vertex Set ◽  
Multipartite Graph ◽  
Complete Multipartite

Let G be a non-trivial finite group, S ? G \ {e} be a set such that if a 2 S, then a-1 ? S and e be the identity element of G. Suppose that Cay(G, S) is the Cayley graph with the vertex set G such that two vertices a and b are adjacent whenever a-1 ? S. An arbitrary graph is called integral whenever all eigenvalues of the adjacency matrix are integers. We say that a group G is Cayley integral simple whenever every connected integral Cayley graph on G is isomorphic to a complete multipartite graph. In this paper we prove that if G is a non-simple group, then G is Cayley integral simple if and only if G ? Zp2 for some prime number p or G ? Z2 x Z2. Moreover, we show that there exist finite non-abelian simple groups which are not Cayley integral simple.

scholarly journals The p-domination number of complete multipartite graphs

2014 ◽  
Vol 06 (04) ◽  
pp. 1450063 ◽  
Author(s):  
You Lu ◽  
Jun-Ming Xu
Keyword(s):  
Positive Integer ◽  
Dominating Set ◽  
Domination Number ◽  
Exact Formula ◽  
Arbitrary Positive Integer ◽  
Minimum Cardinality ◽  
Multipartite Graphs ◽  
Complete Multipartite Graphs ◽  
Complete Multipartite

Let G = (V, E) be a graph and p be a positive integer. A subset S ⊆ V is called a p-dominating set of G if every vertex not in S has at least p neighbors in S. The p-domination number is the minimum cardinality of a p-dominating set in G. This paper establishes an exact formula of the p-domination number of all complete multipartite graphs for arbitrary positive integer p.

Sign in / Sign up

Export Citation Format

Share Document