Journal of Algebraic Structures and Their Applications
Volume:4 Issue: 1, Winter - Spring 2017

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)

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.

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.

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.

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.

The rings considered in this article are commutative with identity which admit at least two maximal idealsý. ýThis article is inspired by the work done on the comaximal ideal graph of a commutative ringý. ýLet R be a ringý. ýWe associate an undirected graph to R denoted by \mathcal{G}(R)ý, ýwhose vertex set is the set of all proper ideals I of R such that I\not\subseteq J(R)ý, ýwhere J(R) is the Jacobson radical of R and distinct vertices I1ý, ýI2are adjacent in \mathcal{G}(R) if and only if I1∩ I2 = I1I2ý. ýThe 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.