Transactions on Combinatorics
Volume:3 Issue: 1, Mar 2014

A recursive-circulant $G(n; d)$ is defined to be a circulant graph with $n$ vertices and jumps of powers of $d$. $G(n; d)$ is vertex-transitive, and has some strong hamiltonian properties. $G(n; d)$ has a recursive structure when $n = cd^m$, $1 leq c < d $ [10]. In this paper, we will find the automorphism group of some classes of recursive-circulant graphs. In particular, we will find that the automorphism group of $G(2^m; 4)$ is isomorphic with the group $D_{2 cdot 2^m}$, the dihedral group of order $2^{m+1}

We denote by LS[N](t; k; v) a large set of t-(v; k; lambda ) designs of size N, which is a partition of all k-subsets of a v-set into N disjoint t-(v; k;lambda )designs and $N={v-t choose k-t}/lambda$. We use the notation N(t; v; k; lambda) as the maximum possible number of mutually disjoint cyclic t-(v; k; lambda)designs. In this paper we give some new bounds for N(2;; 4; 3) and N(2; 31; 4; 2). Within these bounds we present the new large sets LS[9](2; 4;; 29); LS[13](2; 4; 29) and LS[7](2; 4; 31), where their existences were proved before.

For q in {7,8,9,11,13,16}, we consider the primitive actions of L2(q) and use Key-Moori Method 1 as described in [5, 6] to construct designs from the orbits of the point stabilisers and from any union of these orbits. We also use Key-Moori Method 2 (see [8]) to determine the designs from the maximal subgroups and the conjugacy classes of elements of these groups. The incidence matrices of these designs are then used to generate associated binary codes. The full automorphisms of these designs and codes are also determined.

Let G be a simple graph. An independent set is a set of pairwise non-adjacent vertices. The number of vertices in a maximum independent set of G is denoted by (G). In this paper,we characterize graphs G with n vertices and with maximum number of maximum independent setsprovided that (G)  2 or (G)  n 􀀀 3.

Let D be a digraph with skew-adjacency matrix S(D). Then the skew energy of D is defined as the sum of the norms of all eigenvalues of S(D). Denote by On the class of digraphs of order n with no even cycles, and by On,m the class of digraphs in On with m arcs. In this paper, we first give theminimal skew energy digraphs in On and On,m with n − 1  m  3 2 (n − 1). Then we determine the maximal skew energy digraphs in On,n and On,n+1, and in the latter case we assume that n is even.

A watching system in a graph G = (V;E) is a set W = f!1;! 2;: :: ;! kg, where! i = (vi;Zi); vi 2 V and Zi is a subset of closed neighborhood of vi such that the sets LW(v) = f!i: v 2! ig are non-empty and distinct, for any v 2 V. In this paper, we study the watching systems of line graph Kn which is called triangular graph and denoted by T(n). The minimum size of a watching system of G is denoted by! (G). We show that! (T(n)) = d 2n 3 e