Transactions on Combinatorics
Volume:3 Issue: 3, Sep 2014

The sharp upper bounds and the sharp lower bounds of the largest eigenvalues $lambda_1$, the least eigenvalue $lambda_n$, the second largest eigenvalue $lambda_2$, the spread and the separator among all firefly graphs on $n$ vertices are determined.

Let R be a commutative ring with zero-divisor set Z(R). The total graph of R, denoted by T(􀀀(R)),is the simple (undirected) graph with vertex set R, and two distinct vertices are adjacent if their sum is a zero divisor in R. Let T(􀀀(Zn)) be the total graph for the ring of integers modulo n. Minimum zero-sum k-flows for T(􀀀(Zn)) are constructed, in particular, the cases n is even, or n = p^r, n = p^rq^s, where p and q are primes, r and s are positive integers, are considered. Minimum zero-sum k-flows as well as minimum constant-sum k-flows in regular graphs are also investigated.

Let $G$ be a simple graph with vertex set $V(G) = {v_1, v_2,ldots, v_n}$ and $d_i$ the degree of its vertex $v_i$, $i = 1, 2, cdots, n$. Inspired by the Randic matrix and the general Randic index of a graph, we introduce the concept of general Randic matrix $textbf{R}_alpha$ of $G$, which is defined by $(textbf{R}_alpha)_{i,j}=(d_id_j)^alpha$ if $v_i$ and $v_j$ are adjacent, and zero otherwise. Similarly, the general Randi'{c} eigenvalues are the eigenvalues of the general Randi'{c} matrix, the greatest general Randi'{c} eigenvalue is the general Randi'{c} spectral radius of $G$, and the general Randi'{c} energy is the sum of the absolute values of the general Randi'{c} eigenvalues. In this paper, we prove some properties of the general Randic matrix and obtain lower and upper bounds for general Randi'{c} energy, also, we get some lower bounds for general Randi'{c} spectral radius of a connected graph. Moreover, we give a new sharp upper bound for the general Randi'{c} energy when $alpha=-1/2$.[2mm] noindent{bf Keywords:} general Randic matrix, general Randic energy, eigenvalues, spectral radius.

The main result of this paper gives a characterization of association schemes having commutative thin thin residue. This gives a generalization of Ito's Theorem on finite groups for association schemes.

The corona product $Gcirc H$ of two graphs $G$ and $H$ is obtained by taking one copy of $G$ and $|V(G)|$ copies of $H$; and by joining each vertex of the $i$-th copy of $H$ to the $i$-th vertex of $G$, where $1 leq i leq |V(G)|$. In this paper, exact formulas for the eccentric distance sum and the edge revised Szeged indices of the corona product of graphs are presented. We also study the conditions under which the corona product of graphs produces a median graph.

Let L be a lattice with the least element 0. An element x 2 L is a zero divisor if x^y = 0 for some y 2 L = L \ {0}. The set of all zero divisors is denoted by Z(L). We associate a simple graph 􀀀(L) to L with vertex set Z(L) = Z(L) \ {0}, the set of non-zero zero divisors of L and distinct x, y 2 Z(L) are adjacent if and only if x ^ y = 0. In this paper, we obtain certain properties and diameter and girth of the zero divisor graph 􀀀(L). Also we find a dominating set and the domination number of the zero divisor graph 􀀀(L).