Transactions on Combinatorics
Volume:2 Issue: 3, Sep 2013

‎‎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 {em 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 $vin V(G)$ with $f(v)=emptyset$ the condition $bigcup_{uin N(v)}f(u)={1,2}$ is fulfilled, where $N(v)$ is the open neighborhood of $v$. The {em weight} of a 2RDF $f$ is the value $omega(f)=sum_{vin V}|f (v)|$. The {em $2$-rainbow domination number} of a graph $G$, denoted by $gamma_{r2}(G)$, is the minimum weight of a 2RDF of G. The {em 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, $gamma_{r2}(T)le a(T)+1$.

The independence polynomial of a graph G is the polynomial $sum i_kx^k$, where $i_k$ 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.

Bounds for the degree Kirchhoff index of the line and para-line graphs are determined. The special case of regular graphs is analyzed in due detail.

Let G be graph with vertex set V(G) and edge set E(G) and the set A={0,1}. A mapping 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. It depends on the underlying graph G and on its binary labeling; upper and lower bounds for are established. The label energies of some families of graphs are computed.