eng
Yazd University
Algebraic Structures and Their Applications
2382-9761
2423-3447
2017-10-01
4
1
1
18
10.29252/asta.4.1.1
994
مقاله پژوهشی
Small graphs with exactly two non-negative eigenvalues
Tajedin Derikvand
1
Mohammad Reza Oboudi
mr_oboudi@yahoo.com
2
Department of Mathematics, Marvdasht Branch, Islamic Azad University, Marvdasht, Iran
Department of Mathematics, College of Sciences, Shiraz University, Shiraz, 71457-44776, Iran
Let $G$ be a graph with eigenvalues $lambda_1(G)geqcdotsgeqlambda_n(G)$. In this paper we find all simple graphs $G$ such that $G$ has at most twelve vertices and $G$ has exactly two non-negative eigenvalues. In other words we find all graphs $G$ on $n$ vertices such that $nleq12$ and $lambda_1(G)geq0$, $lambda_2(G)geq0$ and $lambda_3(G)0$, $lambda_2(G)>0$ and $lambda_3(G)
http://as.yazd.ac.ir/article_994_708f0d4c89ce19056a0c89be6c5bc68f.pdf
Spectrum of graphs
Eigenvalues of graphs
Graphs with exactly two non-negative eigenvalues
eng
Yazd University
Algebraic Structures and Their Applications
2382-9761
2423-3447
2017-10-01
4
1
19
32
10.29252/asta.4.1.19
1062
مقاله پژوهشی
The Main Eigenvalues of the Undirected Power Graph of a Group
Mehrnoosh Javarsineh
mehrnooshjavar@gmail.com
1
Gholam Hossein Fath-Tabar
gh.fathtabar@gmail.com
2
Department of Pure Mathematics, Faculty of Mathematical Sciences, University of Kashan, Kashan 87317-53153, Iran.
Department of Pure Mathematics, Faculty of Mathematical Sciences, University of Kashan, Kashan 87317-53153, Iran.
The undirected power graph of a finite group $G$, $P(G)$, is a graph with the group elements of $G$ as vertices and two vertices are adjacent if and only if one of them is a power of the other. Let $A$ be an adjacency matrix of $P(G)$. An eigenvalue $lambda$ of $A$ is a main eigenvalue if the eigenspace $epsilon(lambda)$ has an eigenvector $X$ such that $X^{t}jjneq 0$, where $jj$ is the all-one vector. In this paper we want to focus on the power graph of the finite cyclic group $mathbb{Z}_{n}$ and find a condition on n where $P(mathbb{Z}_{n})$ has exactly one main eigenvalue. Then we calculate the number of main eigenvalues of $P(mathbb{Z}_{n})$ where $n$ has a unique prime decomposition $n = p^{r} p_2$. We also formulate a conjecture on the number of the main eigenvalues of $P(mathbb{Z}_{n})$ for an arbitrary positive integer $n$.
http://as.yazd.ac.ir/article_1062_147393a1dbdb1178bfcbf7af5101e35f.pdf
Power graph
Main eigenvalue
Cyclic group
Prime divisor
eng
Yazd University
Algebraic Structures and Their Applications
2382-9761
2423-3447
2017-10-01
4
1
33
42
10.29252/asta.4.1.33
1066
مقاله پژوهشی
On the Edge-Difference and Edge-Sum Chromatic Sum of the Simple Graphs
S. Rahimi Sharbaf
1
Kh. Erfani
2
School of Mathematical Science, Shahrood University of Technology, Shahrood, Iran.
School of Mathematical Science, Shahrood University of Technology, Shahrood, Iran.
For a coloring $c$ of a graph $G$, the edge-difference coloring sum and edge-sum coloring sum with respect to the coloring $c$ are respectively $sum_c D(G)=sum |c(a)-c(b)|$ and $sum_s S(G)=sum (c(a)+c(b))$, where the summations are taken over all edges $abin E(G)$. The edge-difference chromatic sum, denoted by $sum D(G)$, and the edge-sum chromatic sum, denoted by $sum S(G)$, are respectively the minimum possible values of $sum_c D(G)$ and $sum_c S(G)$, where the minimums are taken over all proper coloring of $c$. In this work, we study the edge-difference chromatic sum and the edge-sum chromatic sum of graphs. In this regard, we present some necessary conditions for the existence of homomorphism between two graphs. Moreover, some upper and lower bounds for these parameters in terms of the fractional chromatic number are introduced as well.
http://as.yazd.ac.ir/article_1066_632a2cc0dfa501dfe96f93a086cd2645.pdf
edge-difference chromatic sum
edge-sum chromatic sum
graph homomorphism
Kneser graph
fractional chromatic number
eng
Yazd University
Algebraic Structures and Their Applications
2382-9761
2423-3447
2018-02-15
4
1
43
50
10.29252/asta.4.1.43
1070
مقاله پژوهشی
On two-dimensional Cayley graphs
Ali Behtoei
a.behtoei@sci.ikiu.ac.ir
1
yasser Golkhandy Pour
y.golkhandypour@edu.ikiu.ac.ir
2
Department of Mathematics Imam Khomeini International University, P.O, Box 34149-16818 Qazvin, Iran
Department of Mathematics, Faculty of sciences, Imam Khomeini International University, Qazvin, Iran
A subset W of the vertices of a graph G is a resolving set for G when for each pair of distinct vertices u,v in V (G) there exists w in W such that d(u,w)≠d(v,w). The cardinality of a minimum resolving set for G is the metric dimension of G. This concept has applications in many diverse areas including network discovery, robot navigation, image processing, combinatorial search and optimization. The problem of finding metric dimension is NP-complete for general graphs but the metric dimension of trees can be obtained using a polynomial time algorithm. In this paper, we investigate the metric dimension of Cayley graphs on dihedral groups and we characterize a family of them.
http://as.yazd.ac.ir/article_1070_f86df4fea8c47ba4362a6a18ef53a53a.pdf
Resolving set
Metric dimension
Cayley graph
Dihedral group
eng
Yazd University
Algebraic Structures and Their Applications
2382-9761
2423-3447
2018-03-15
4
1
51
56
10.29252/asta.4.1.51
1089
مقاله پژوهشی
D-Spectrum and D-Energy of Complements of Iterated Line Graphs of Regular Graphs
Indulal Gopalapillai
indulalgopal@gmail.com
1
Department of Mathematics, St.Aloysius College, Edathua, Alappuzha, India
The D-eigenvalues {µ1,…,µp} of a graph G are the eigenvalues of its distance matrix D and form its D-spectrum. The D-energy, ED(G) of G is given by ED (G) =∑i=1p |µi|. Two non cospectral graphs with respect to D are said to be D-equi energetic if they have the same D-energy. In this paper we show that if G is an r-regular graph on p vertices with 2r ≤ p - 1, then the complements of iterated line graphs of G are of diameter 2 and that ED(overline{Lk(G)}), k≥2 depends only on p and r. This result leads to the construction of regular D-equi energetic pair of graphs.
http://as.yazd.ac.ir/article_1089_78d7d575d5ef4cc6f63444bee6a8aafb.pdf
Distance spectrum
Distance energy
Line graphs
eng
Yazd University
Algebraic Structures and Their Applications
2382-9761
2423-3447
2018-04-22
4
1
57
76
10.29252/asta.4.1.57
1123
مقاله پژوهشی
A note on a graph related to the comaximal ideal graph of a commutative ring
Subramanian Visweswaran
s_visweswaran2006@yahoo.co.in
1
Jaydeep Parejiya
2
Department of Mathematics, Saurashtra University, Rajkot, India.
Department of Mathematics, Saurashtra University, Rajkot, India.
The rings considered in this article are commutative with identity which admit at least two maximal ideals. This article is inspired by the work done on the comaximal ideal graph of a commutative ring. Let R be a ring. We associate an undirected graph to R denoted by mathcal{G}(R), whose vertex set is the set of all proper ideals I of R such that Inotsubseteq J(R), where J(R) is the Jacobson radical of R and distinct vertices I1, I2are adjacent in mathcal{G}(R) if and only if I1∩ I2 = I1I2. The aim of this article is to study the interplay between the graph-theoretic properties of mathcal{G}(R) and the ring-theoretic properties of R.
http://as.yazd.ac.ir/article_1123_02c4c66101785f6e69330945831e4fce.pdf
Comaximal ideal graph of a commutative ring
complete graph
von Neumann regular ring
bipartite graph
clique number