Transactions on Combinatorics
Volume:1 Issue: 2, Jun 2012

S. Kim، et al، have been analyzed the girth of some algebraically structured quasi-cyclic (QC) low-density parity-check (LDPC) codes، i. e. Tanner (3،5) of length 5p، where p is a prime of the form 15m+1. In this paper، by extension this method to Tanner (3،7) codes of length 7p، where p is a prime of the form 21m+ 1، the girth values of Tanner (3،7) codes will be derived. As an advantage، the rate of Tanner (3،7) codes is about 0. 17 more than the rate of Tanner (3،5) codes.

A dominating set $D subseteq V$ of a graph $G = (V،E) $ is said to be a connected cototal dominating set if $langle D rangle$ is connected and $langle V-D rangle neq phi$، contains no isolated vertices. A connected cototal dominating set is said to be minimal if no proper subset of $D$ is connected cototal dominating set. The connected cototal domination number $gamma_ {ccl} (G) $ of $G$ is the minimum cardinality of a minimal connected cototal dominating set of $G$. In this paper، we begin an investigation of connected cototal domination number and obtain some interesting results.

Let G be a connected graph with vertex set V (G). The degree resistance distance of G is defined as the sum over all pairs of vertices of the terms [d (u) +d (v)] R (u،v)، where d (u) is the degree of vertex u، and R (u،v) denotes the resistance distance between u and v. In this paper، we characterize n-vertex unicyclic graphs having minimum and second minimum degree resistance distance.

The non commuting graph of a non-abelian finite group $G$ is defined as follows: its vertex set is $G-Z (G) $ and two distinct vertices $x$ and $y$ are joined by an edge if and only if the commutator of $x$ and $y$ is not the identity. In this paper we prove some new results about this graph. In particular we will give a new proof of theorem 3. 24 of [2]. We also prove that if $G_1$، $G_2$،…، $G_n$ are finite groups such that $Z (G_i) =1$ for $i=1،2،…،n$ and they are characterizable by non commuting graph، then $G_1times…times G_n$ is characterizable by non commuting graph.

Let G be a simple graph of order n. We consider the independence polynomial and the domination polynomial of a graph G. The value of a graph polynomial at a specific point can give sometimes a very surprising information about the structure of the graph. In this paper we investigate independence and domination polynomial at -1 and 1.

In this paper we introduce the concept of order difference interval graph ¡ODI (G) of a group G. It is a graph ¡ODI (G) with V (¡ODI (G)) = G and two vertices a and b are adjacent in ¡ODI (G) if and only if o (b) − o (a) ∈ [o (a)، o (b)]. Without loss of generality، we assume that o (a) ≤ o (b). In this paper we obtain several properties of ¡ODI (G)، upper bounds on the number of edges of ¡ODI (G) and determine those groups whose order difference interval graph is isomorphic to a complete multipartite graph.