Small graphs with exactly two non-negative eigenvalues
Tajedin
Derikvand
Department of Mathematics, Marvdasht Branch, Islamic Azad University, Marvdasht, Iran
author
Mohammad Reza
Oboudi
Department of Mathematics, College of Sciences, Shiraz University, Shiraz, 71457-44776, Iran
author
text
article
2017
eng
Let $G$ be a graph with eigenvalues $\lambda_1(G)\geq\cdots\geq\lambda_n(G)$. 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 $n\leq12$ and $\lambda_1(G)\geq0$, $\lambda_2(G)\geq0$ and $\lambda_3(G)<0$. We obtain that there are exactly $1575$ connected graphs $G$ on $n\leq12$ vertices with $\lambda_1(G)>0$, $\lambda_2(G)>0$ and $\lambda_3(G)<0$. We find that among these $1575$ graphs there are just two integral graphs.
Algebraic Structures and Their Applications
Yazd University
2382-9761
4
v.
1
no.
2017
1
18
http://as.yazd.ac.ir/article_994_708f0d4c89ce19056a0c89be6c5bc68f.pdf
dx.doi.org/10.29252/asta.4.1.1
The Main Eigenvalues of the Undirected Power Graph of a Group
Mehrnoosh
Javarsineh
Department of Pure Mathematics, Faculty of Mathematical Sciences, University of Kashan, Kashan 87317-53153, Iran.
author
Gholam Hossein
Fath-Tabar
Department of Pure Mathematics, Faculty of Mathematical Sciences, University of Kashan, Kashan 87317-53153, Iran.
author
text
article
2017
eng
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 $X^{t}\jj\neq 0$, 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 $n$.
Algebraic Structures and Their Applications
Yazd University
2382-9761
4
v.
1
no.
2017
19
32
http://as.yazd.ac.ir/article_1062_147393a1dbdb1178bfcbf7af5101e35f.pdf
dx.doi.org/10.29252/asta.4.1.19
On the Edge-Difference and Edge-Sum Chromatic Sum of the Simple Graphs
S.
Rahimi Sharbaf
School of Mathematical Science, Shahrood University of Technology, Shahrood, Iran.
author
Kh.
Erfani
School of Mathematical Science, Shahrood University of Technology, Shahrood, Iran.
author
text
article
2017
eng
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 $ab\in E(G)$. 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.
Algebraic Structures and Their Applications
Yazd University
2382-9761
4
v.
1
no.
2017
33
42
http://as.yazd.ac.ir/article_1066_632a2cc0dfa501dfe96f93a086cd2645.pdf
dx.doi.org/10.29252/asta.4.1.33
On two-dimensional Cayley graphs
Ali
Behtoei
Department of Mathematics
Imam Khomeini International University,
P.O, Box 34149-16818
Qazvin, Iran
author
yasser
Golkhandy Pour
Department of Mathematics, Faculty of sciences, Imam Khomeini International University, Qazvin, Iran
author
text
article
2018
eng
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.
Algebraic Structures and Their Applications
Yazd University
2382-9761
4
v.
1
no.
2018
43
50
http://as.yazd.ac.ir/article_1070_f86df4fea8c47ba4362a6a18ef53a53a.pdf
dx.doi.org/10.29252/asta.4.1.43
D-Spectrum and D-Energy of Complements of Iterated Line Graphs of Regular Graphs
Indulal
Gopalapillai
Department of Mathematics, St.Aloysius College, Edathua, Alappuzha, India
author
text
article
2018
eng
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.
Algebraic Structures and Their Applications
Yazd University
2382-9761
4
v.
1
no.
2018
51
56
http://as.yazd.ac.ir/article_1089_78d7d575d5ef4cc6f63444bee6a8aafb.pdf
dx.doi.org/10.29252/asta.4.1.51
A note on a graph related to the comaximal ideal graph of a commutative ring
Subramanian
Visweswaran
Department of Mathematics, Saurashtra University, Rajkot, India.
author
Jaydeep
Parejiya
Department of Mathematics, Saurashtra University, Rajkot, India.
author
text
article
2018
eng
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 I\not\subseteq 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.
Algebraic Structures and Their Applications
Yazd University
2382-9761
4
v.
1
no.
2018
57
76
http://as.yazd.ac.ir/article_1123_02c4c66101785f6e69330945831e4fce.pdf
dx.doi.org/10.29252/asta.4.1.57