### فهرست مطالب • Volume:4 Issue: 1, Winter - Spring 2017
• تاریخ انتشار: 1395/12/20
• تعداد عناوین: 6
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• Tajedin Derikvand, Mohammad Reza Oboudi * Pages 1-18
Let G be a graph with eigenvalues 1(G) n(G). In this paper we nd all simple graphs G such that G has at most twelve vertices and G has exactly two non-negative eigenvalues. In other words we nd all graphs G on n vertices such that n  12 and 1(G)  0, 2(G)  0 and 3(G) 0, 2(G) > 0 and 3(G)
Keywords: Spectrum of graphs, Eigenvalues of graphs, Graphs with exactly two non, negative eigenvalues
• Mehrnoosh Javarsineh, Gholam Hossein Fath-Tabar * Pages 19-32
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 λ of A is a main eigenvalue if the eigenspace ϵ(λ) has an eigenvector X such that X t \jj≠0, where \jj is the all-one vector. In this paper we want to focus on the power graph of the finite cyclic group Z n and find a condition on n where P(Z n ) has exactly one main eigenvalue. Then we calculate the number of main eigenvalues of P(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(Z n ) for an arbitrary positive integer n.
Keywords: Power graph, Main eigenvalue, Cyclic group, Prime divisor
• S. Rahimi Sharbaf, Kh. Erfani * Pages 33-42
ýFor a coloring c of a graph G ý, ýthe edge-difference coloring sum and edge-sum coloring sum with respect to the coloring c are respectivelyý ý∑ c D(G)=∑|c(a)−c(b)| and ∑sS(G)=∑(c(a)(b))ý, ýwhere the summations are taken over all edges ab∈E(G)ý.
ýThe edge-difference chromatic sumý, ýdenoted by ∑D(G) ý, ýand the edge-sum chromatic sumý, ýdenoted by ∑S(G) ý, ýare respectively the minimum possible valuesý ýof ∑ c D(G) and ∑ c S(G)ý, ýwhere the minimums are taken over all proper coloring of c ý.ýIn this worký, ýwe study the edge-difference chromatic sum and the edge-sum chromatic sum of graphsý. ýIn this regardý,ýwe present some necessary conditions for the existence of homomorphism between two graphsý. ýMoreoverý, ýsome upper and lower bounds for these parameters in terms of the fractional chromatic number are introducedý ýas wellý.
Keywords: ?edge, difference chromatic sum?, ?edge, sum chromatic sum?, ?graph homomorphism?, ?Kneser graph?, ?fractional chromatic number
• Ali Behtoei *, Yasser Golkhandy Pour Pages 43-50
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.
Keywords: Resolving set, Metric dimension, Cayley graph, Dihedral group
• Indulal Gopalapillai * Pages 51-56
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.
Keywords: Distance spectrum, Distance energy, Line graphs
• Subramanian Visweswaran *, Jaydeep Parejiya Pages 57-76