In this paper we introduce mixed unitary Cayley graph $M_{n}$ $(n>1)$ and compute its eigenvalues. We also compute the energy of $M_{n}$ for some $n$.

In this paper we introduce mixed unitary Cayley graph $M_{n}$ $(n>1)$ and compute its eigenvalues. We also compute the energy of $M_{n}$ for some $n$.

In this paper, a recursive algorithm is presented to generate some exponent matrices which correspond to Tanner graphs with girth at least 6. For a $J times L$ 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 $m geq Q(E)$. The results show that the exponent matrices constructed with our recursive algorithm have smaller lower-bound than the ones proposed recently with girth 6.

The Gutman index and degree distance of a connected graph $G$ are defined as begin{eqnarray*} textrm{Gut}(G)=sum_{{u,v}subseteq V(G)}d(u)d(v)d_G(u,v), end{eqnarray*} and begin{eqnarray*} DD(G)=sum_{{u,v}subseteq V(G)}(d(u)+d(v))d_G(u,v), end{eqnarray*} respectively, where $d(u)$ is the degree of vertex $u$ and $d_G(u,v)$ is the distance between vertices $u$ and $v$. In this paper, through a recurrence equation for the Wiener index, we study the first two moments of the Gutman index and degree distance of increasing trees.

The divisibility graph $mathscr{D}(G)$ for a finite group $G$ is a graph with vertex set $cs(G)setminus{1}$ where $cs(G)$ is the set of conjugacy class sizes of $G$. Two vertices $a$ and $b$ are adjacent whenever $a$ divides $b$ or $b$ divides $a$. In this paper we will find the number of connected components of $mathscr{D}(G)$ where $G$ is a simple Zassenhaus group or an sporadic simple group.

Two Latin squares of order $n$ are orthogonal if in their superposition, each of the $n^{2}$ ordered pairs of symbols occurs exactly once. Colbourn, Zhang and Zhu, in a series of papers, determined the integers $r$ for which there exist a pair of Latin squares of order $n$ having exactly $r$ different ordered pairs in their superposition. Dukes and Howell defined the same problem for Latin squares of different orders $n$ and $n+k$. They obtained a non-trivial lower bound for $r$ and solved the problem for $k geq frac{2n}{3} $. Here for $k < frac{2n}{3}$, some constructions are shown to realize many values of $r$ and for small cases $(3leq n leq 6)$, the problem has been solved.