Let \(G\) be a simple graph. A set \(D\) of vertices of \(G\) is dominating if every vertex not in \(D\) is adjacent to some vertex in \(D\). A set \(M\) of edges of \(G\) is called independent, or a matching, if no two edges of \(M\) are adjacent in \(G\). The domination number \(\gamma(G)\) is the minimum order of a dominating set in \(G\). The edge independence number \(\alpha_0(G)\) is the maximum size of a matching in \(G\). If \(G\) has no isolated vertices, then the inequality \(\gamma(G) \leq \alpha_0(G)\) holds. In this paper we characterize regular graphs, unicyclic graphs, block graphs, and locally connected graphs for which \(\gamma(G) = \alpha_0(G)\).

Let \(n \geq 1\) be an integer and let \(G\) be a graph of order \(p\). A set \(I_n\) of vertices of \(G\) is \(n\)-independent if the distance between every two vertices of \(I_n\) is at least \(n+1\). Furthermore, \(I_n\) is defined to be an \(n\)-independent dominating set of \(G\) if \(I_n\) is an \(n\)-independent set in \(G\) and every vertex in \(V(G) – I_nv is at distance at most \(n\) from some vertex in \(I_n\). The \(n\)-independent domination number, \(i_n(G)\), is the minimum cardinality among all \(n\)-independent dominating sets of \(G\). Hence \(i_n(G) = i(G)\) where \(i(G)\) is the independent domination number of \(G\). We establish the existence of a connected graph \(G\) every spanning tree \(T\) of which is such that \(i_n(T) < i_n(G)\). For \(n \in \{1,2\}\) we show that, for any tree \(T\) and any tree \(T’\) obtained from \(T\) by joining a new vertex to some vertex of \(T\), we have \(i_n(T) \geq i_n(T’)\). However, we show that this is not true for \(n \geq 3\). We show that the decision problem corresponding to the problem of computing \(i_n(G)\) is NP-complete, even when restricted to bipartite graphs. Finally, we obtain a sharp lower bound on \(i_n(G)\) for a graph \(G\).

In this paper, we consider symmetric and skew equivalence of Hadamard matrices of order \(28\) and present some computational results and some applications.

We consider a linear model for the comparison \(V \geq 2\) treatments (or one treatment at \(V\) levels) in a completely randomized statistical setup, making \(r\) (the replication number) observations per treatment level in the presence of \(K\) continuous covariates with values on the \(K\)-cube. The main interest is restricted to cyclic designs characterized by the property that the allocation matrix of each treatment level is obtained through cyclic permutation of the columns of the allocation matrix of the first treatment level. The \(D\)-optimality criterion is used for estimating all the parameters of this model.

By studying the nonperiodic autocorrelation function of circulant matrices, we develop an exhaustive algorithm for constructing \(D\)-optimal cyclic designs with even replication number. We apply this algorithm for \(r = 4, 16 \leq V \leq 24, N=rV \equiv 0 \mod 4\), for \(r=6, 12\leq V \leq 24, N =rV \equiv 0 \mod 4\), for \(r =6, V =m.n, m\) is a prime, \(N =rV \equiv 2 \mod 4\) and the corresponding cyclic designs are given.

New sufficient conditions for equality of edge-connectivity and minimum degree of graphs are presented, including those of Chartrand, Lesniak, Plesnik, Plesník, and Ždmán, and Volkmann.

We show that \((81, 16, 3)\)-block designs have no involutionary automorphisms that fix just \(13\) points. Since the nonexistence of \((81, 16, 3)\)-designs with involutionary automorphism fixing \(17\) points has already been proved, it follows that any involution that an \((81, 16, 3)\)-design may have must fix just \(9\) points.

In this paper we construct a latin \((n \times n \times (n-d))\)-parallelepiped that cannot be extended to a latin cube of order \(n\) for any pair of integers \(d, n\) where \(d \geq 3\) and \(n \geq 2d+1\). For \(d = 2\), it is similar to the construction already known.

In this note the numbers of common triples in two simple balanced ternary designs with block size \(3\), index \(3\) and \(p_2 = 3\) are determined.

The class of parity graphs, those in which the cardinality of every maximal independent subset of vertices has the same parity, contains the well-covered graphs and arose in connection with the PSPACE-complete game “Generalized Kayles”. In 1983 [5] we characterized parity graphs of girth 8 or more. This is extended to a characterization of the parity graphs of girth greater than 5. We deduce that these graphs can be recognized in polynomial time.