Algebraic Structures and Their ApplicationsAlgebraic Structures and Their Applications
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Feed provided by Algebraic Structures and Their Applications. Click to visit.Small graphs with exactly two non-negative eigenvalues
http://as.yazd.ac.ir/article_994_212.html
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 $lambda_3(G)<0$. We obtain that there are exactly $1575$ connected graphs $G$ on $nleq12$ 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.Sat, 30 Sep 2017 20:30:00 +0100The Main Eigenvalues of the Undirected Power Graph of a Group
http://as.yazd.ac.ir/article_1062_212.html
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 $n$.Sat, 30 Sep 2017 20:30:00 +0100On the Edge-Difference and Edge-Sum Chromatic Sum of the Simple Graphs
http://as.yazd.ac.ir/article_1066_212.html
‎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.Sat, 30 Sep 2017 20:30:00 +0100On two-dimensional Cayley graphs
http://as.yazd.ac.ir/article_1070_212.html
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. Wed, 14 Feb 2018 20:30:00 +0100