TRANSACTIONS ON COMBINATORICS
Year:2013 | Volume:2 | Issue:3|Papers Count:6

In this paper, the type-II matrices on (negative) Latin square graphs are considered and it is proved that, under certain conditions, the Nomura algebras of such type-II matrices are trivial. In addition, we construct type-II matrices on doubly regular tournaments and show that the Nomura algebras of such matrices are also trivial.

An equitable domination has interesting application in the context of social networks. In a network, nodes with nearly equal capacity may interact with each other in a better way. In the society persons with nearly equal status, tend to be friendly. In this paper, we introduce new variant of equitable domination of a graph. Basic properties and some interesting results have been obtained.

A 2-rainbow dominating function (2RDF) of a graph G is a function f from the vertex set V (G) to the set of all subsets of the set {1, 2} such that for any vertex vÎV (G) with f (v) = f the condition UuÎN (v) f (u) = {1, 2} is fulfilled, where N (v) is the open neighborhood of v. The weight of a 2RDF f is the value w (f) = SvÎV |f (v) |. The 2-rainbow domination number of a graph G, denoted by yr2 (G), is the minimum weight of a 2RDF of G. The annihilation number a (G) is the largest integer k such that the sum of the first k terms of the non-decreasing degree sequence of G is at most the number of edges in G. In this paper, we prove that for any tree T with at least two vertices, yr2 (T) £ a (T) +1.

The independence polynomial of a graph G is the polynomial S ikxk, where ik denote the number of independent sets of cardinality k in G. In this paper we study unimodality problem for the independence polynomial of certain classes of graphs.

The degree Kirchhoff index of a connected graph G is defined as the sum of the terms di dj rij over all pairs of vertices, where di is the degree of the i-th vertex, and rij the resistance distance between the i-th and j-th vertex of G. Bounds for the degree Kirchhoff index of the line and para-line graphs are determined. The special case of regular graphs is analyzed.

Let G be a graph with vertex set V (G) and edge set X (G) and consider the set A = {0, 1}. A mapping l: V (G)®A is called binary vertex labeling of G and l (v) is called the label of the vertex v under l. In this paper we introduce a new kind of graph energy for the binary labeled graph, the labeled graph energy El (G). It depends on the underlying graph G and on its binary labeling, upper and lower bounds for El (G) are established. The labeled energies of a number of well known and much studied families of graphs are computed.