Transactions on Combinatorics
Volume:5 Issue: 2, Jun 2016

ýIn this paper we introduce mixed unitary Cayley graph Mn (n>1) and compute its eigenvaluesý. ýWe also compute the energy ofý Mn for some n.

In this paper, a recursive algorithm is presented to generate some exponent matrices whichcorrespond to Tanner graphs with girth at least 6. For a JxL exponent matrix E, the lower bound Q(E) is obtained explicitly such that (J;L) QC LDPC codes with girth at least 6 exist for any circulant permutation matrix (CPM) size m  Q(E). The results show that the exponent matrices constructed with our recursive algorithm have smaller lower-bound than the ones proposed recently with girth 6.

ýThe Gutman index and degree distance of a connected graph G are defined asý ,ý ýýGut(G)=∑{u,v}⊆V(G)d(u)d(v)dG(u,v)ý,ýandýý,ýýýDD(G)=∑{u,v}⊆V(G)(d(u)(v))dG(u,v)ý,ý ý ýrespectivelyý, ýwhereý d(u) is the degree of vertex u and dG(u,v) is the distance between vertices u and v ý. ýIn this paperý, ýthrough a recurrence equation for the Wiener indexý, ýwe study the first twoý ýmoments of the Gutman index and degree distance of increasingý ýtreesý.

The divisibility graph D(G) for a finite group G is a graph with vertex set cs(G)∖{1}ý ýwhere cs(G) is the set of conjugacy class sizes of G ý. ýTwo vertices a and b are adjacent whenever a dividesý ýb or b divides a ý. ýIn this paper we will find the number of connected components of D(G) where G is aý ýsimple Zassenhaus group or an sporadic simple groupý.

Two Latin squares of order n are orthogonal if in their superposition, each of the n2 ordered pairs of symbols occurs exactly once. Colbourn, Zhang and Zhu, in a seriesof papers, determined the integers r for which there exist a pair of Latin squares oforder n having exactly r different ordered pairs in their superposition. Dukes and Howell defined thesameproblem for Latin squares of different orders n and n.They obtained a non trivial lower bound for r and solved the problem for kgeq 2n/3.Here for k