Let $g_k(n)=\sum _{\underset{\_}{v}\in C_k(n)}(\genfrac{}{}{0ex}{}{n}{v})2^{v_1{v}_2+v_2{v}_3+v_3{v}_4+\dots +v_{k-1}{v}_k}$ where $C_k(n)$ denote the set of $k$-compositions of $n$. We show that

1. $g_k(n+p-1)\equiv g_k(n)(\mathrm{mod}p)$ for all $k,n\ge 1$, prime $p$;
2. $g_k(n)$ is a polynomial in $k$ of degree $n$ for $k\ge n+1$;

and, moreover, that these properties hold for wider classes of functions which are sums involving multinomial coefficients.

A connected graph $G$ is unicentered if $G$ has exactly one central vertex. It is proved that for integers $r$ and $d$ with $1\le r<d\le 2r$, there exists a unicentered graph $G$ such that rad$(G)=r$ and diam$(G)=d$. Also, it is shown that for any two graphs $F$ and $G$ with rad$(F)=n\ge 4$ and a positive integer $d$ ($4\le d\le n$), there exists a connected graph $H$ with diam$(H)=d$ such that the periphery and the center of $H$ are isomorphic to $F$ and $G$, respectively.

In this paper we obtain some inequalities on the existence of balanced arrays ($B$-arrays) of strength four in terms of its parameter by using Minkowski’s inequality.

Let $q$ be a prime power, $F_{q^2}$ the finite field with $q^2$ elements, $U_n(F_{q^2})$ the finite unitary group of degree $n$ over $F_{q^2}$, and $U{V}_n(F_{q^2})$ the $n$-dimensional unitary geometry over $F_{q^2}$. It is proven that the subgroup consisting of the elements of $U_n(F_{q^2})$ which fix a given $(m,s)$-type subspace of $U{V}_n(F_{q^2})$, acts transitively on some subsets of subspaces of $U{V}_n(F_{q^2})$. This observation gives rise to a number of Partially Balanced Incomplete Block Designs (PBIBD’s).

There are two criteria for optimality of weighing designs. One, which has been widely studied, is that the determinant of $X{X}^T$ should be maximal, where $X$ is the weighing matrix. The other is that the trace of $(X{X}^T{)}^{-1}$ should be minimal. We examine the second criterion. It is shown that Hadamard matrices, when they exist, are optimal with regard to the second criterion, just as they are for the first one.

In 1988, Sarvate and Seberry introduced a new method of construction for the family of weighing matrices $W(n^2(n-1),n^2)$, where $n$ is a prime power. We generalize this result, replacing the condition on $n$ with the weaker assumption that a generalized Hadamard matrix $GH(n;G)$ exists with $|G|=n$, and give conditions under which an analogous construction works for $|G|<n$. We generalize a related construction for a $W(13,9)$, also given by Sarvate and Seberry, producing a whole new class. We build further on these ideas to construct several other classes of weighing matrices.

For an integer $\ell \ge 2$, the $\ell$-connectivity ($\ell$-edge-connectivity) of a graph $G$ of order $p$ is the minimum number of vertices (edges) that need to be deleted from $G$ to produce a disconnected graph with at least $\ell$ components or a graph with at most $\ell -1$ vertices. Let $G$ be a graph of order $p$ and $\ell$-connectivity ${\kappa }_{\ell }$. For each $k\in \{0,1,\dots ,{\kappa }_{\ell }\}$, let $s_k$ be the minimum $\ell$-edge-connectivity among all graphs obtained from $G$ by deleting $k$ vertices from $G$. Then $f_{\ell }=\{(0,s_0),\dots ,({\kappa }_{\ell },s_{{\kappa }_{\ell }})\}$ is the $\ell$-connectivity function of $G$. The $\ell$-connectivity functions of complete multipartite graphs and caterpillars are determined.

An infinite class of graphs is constructed to demonstrate that the difference between the independent domination number and the domination number of $3$-connected cubic graphs may be arbitrarily large.

The domination number $\gamma (G)$ and the total domination number ${\gamma }_t(G)$ of a graph are generalized to the $K_n$-domination number ${\gamma }_{k_n}(G)$ and the total $K_n$-domination number ${\gamma }_{K_n}^t(G)$ for $n\ge 2$, where $\gamma (G)={\gamma }_{K_2}(G)$ and ${\gamma }_t(G)={\gamma }_{K_2}^t(G)$. A nondecreasing sequence $a_2,a_3,\dots ,a_m$ of positive integers is said to be a $K_n$-domination (total $K_n$-domination, respectively) sequence if it can be realized as the sequence of generalized domination (total domination, respectively) numbers ${\gamma }_{K_2}(G),{\gamma }_{K_3}(G),\dots ,{\gamma }_{K_m}(G)$ (${\gamma }_{K_2}^t(G),{\gamma }_{K_3}^t(G),\dots ,{\gamma }_{K_m}^t(G)$, respectively) of some graph $G$. It is shown that every nondecreasing sequence $a_2,a_3,\dots ,a_m$ of positive integers is a $K_n$-domination sequence (total $K_n$-domination sequence, if $a_2\ge 2$, respectively). Further, the minimum order of a graph $G$ with $a_2,a_3,\dots ,a_m$ as a $K_n$-domination sequence is determined. $K_n$-connectivity is defined and for $a_2\ge 2$ we establish the existence of a $K_m$-connected graph with $a_2,a_3,\dots ,a_m$ as a $K_n$-domination sequence for every nondecreasing sequence $a_2,a_3,\dots ,a_m$ of positive integers.