Yazd University
Algebraic Structures and Their Applications
2382-9761
4
1
2017
10
01
Small graphs with exactly two non-negative eigenvalues
1
18
EN
Tajedin
Derikvand
Department of Mathematics, Marvdasht Branch, Islamic Azad University, Marvdasht, Iran
Mohammad Reza
Oboudi
Department of Mathematics, College of Sciences, Shiraz University, Shiraz, 71457-44776, Iran
mr_oboudi@yahoo.com
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)
Spectrum of graphs,Eigenvalues of graphs,Graphs with exactly two non-negative eigenvalues
http://as.yazd.ac.ir/article_994.html
http://as.yazd.ac.ir/article_994_82f88883e4077144c542caa72f4f4bba.pdf
Yazd University
Algebraic Structures and Their Applications
2382-9761
4
1
2017
10
01
The Main Eigenvalues of the Undirected Power Graph of a Group
19
32
EN
Mehrnoosh
Javarsineh
Department of Pure Mathematics, Faculty of Mathematical Sciences, University of Kashan, Kashan 87317-53153, Iran.
mehrnooshjavar@gmail.com
Gholam Hossein
Fath-Tabar
Department of Pure Mathematics, Faculty of Mathematical Sciences, University of Kashan, Kashan 87317-53153, Iran.
gh.fathtabar@gmail.com
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 $n$.
Power graph,Main eigenvalue,Cyclic group,Prime divisor
http://as.yazd.ac.ir/article_1062.html
http://as.yazd.ac.ir/article_1062_eaba7fdbeeddfd92b4fc4c6a379f104a.pdf
Yazd University
Algebraic Structures and Their Applications
2382-9761
4
1
2017
10
01
On the Edge-Difference and Edge-Sum Chromatic Sum of the Simple Graphs
33
42
EN
S.
Rahimi Sharbaf
School of Mathematical Science, Shahrood University of Technology, Shahrood, Iran.
Kh.
Erfani
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.
edge-difference chromatic sum,edge-sum chromatic sum,graph homomorphism,Kneser graph,fractional chromatic number
http://as.yazd.ac.ir/article_1066.html
http://as.yazd.ac.ir/article_1066_fcffe8e641c12d751a41fd8a5f7fdbdf.pdf
Yazd University
Algebraic Structures and Their Applications
2382-9761
4
1
2018
02
15
On two-dimensional Cayley graphs
43
50
EN
Ali
Behtoei
Department of Mathematics
Imam Khomeini International University,
P.O, Box 34149-16818
Qazvin, Iran
a.behtoei@sci.ikiu.ac.ir
yasser
Golkhandy Pour
Department of Mathematics, Faculty of sciences, Imam Khomeini International University, Qazvin, Iran
y.golkhandypour@edu.ikiu.ac.ir
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.
Resolving set,Metric dimension,Cayley graph,Dihedral group
http://as.yazd.ac.ir/article_1070.html
http://as.yazd.ac.ir/article_1070_c39eda19327abe5934a0a39300f17643.pdf
Yazd University
Algebraic Structures and Their Applications
2382-9761
4
1
2018
03
15
D-Spectrum and D-Energy of Complements of Iterated Line Graphs of Regular Graphs
51
56
EN
Indulal
Gopalapillai
Department of Mathematics, St.Aloysius College, Edathua, Alappuzha
indulalgopal@gmail.com
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.
Distance spectrum,Distance energy,Line graphs
http://as.yazd.ac.ir/article_1089.html
http://as.yazd.ac.ir/article_1089_b58cf3e42ffdc51ea7d42d366308c9af.pdf