A vertex \(w\) in a di(graph) \(G\) is said to resolve a pair \(u, v\) of vertices of \(G\) if the distance from \(u\) to \(w\) does not equal the distance from \(v\) to \(w\). A set \(S\) of vertices of \(G\) is a resolving set for \(G\) if every pair of vertices of \(G\) is resolved by some vertex of \(S\). The smallest cardinality of a resolving set for \(G\), denoted by \(dim(G)\), is called the metric dimension for \(G\).

We show that if \(G\) is the Cayley digraph \(Cay(\Delta : \Gamma)\) where \(\Delta = \{ (1, 0, 0), (0, 1, 0), (0, 0, 1) \}\) and \(\Gamma =\mathbb{Z}_m \oplus \mathbb{Z}_n \oplus \mathbb{Z}_k\) with \(m \leq n \leq k\), then \(dim(G) = n\) if \(m < n\) and improve known upper bounds if \(m = n\). We use these results to establish improved upper bounds for the metric dimension of Cayley digraphs of abelian groups that are expressed as a direct product of four or more cyclic groups. Lower bounds for Cayley digraphs of groups that are multiple copies of \(\mathbb{Z}_n\) are established.

In this paper, the unimodality of \((r,\beta)\)-Stirling numbers and certain asymptotic approximation of \((r,\beta)\)-Bell numbers are established. Together with these results and the most general form of Central Limit Theorem, viz. Bounded Variance Normal Convergence Criterion, the \((r,\beta)\)-Stirling numbers are shown to be asymptotically normal.

The graph \(\mathcal{R}(d)\) of realizations of \(d\) is a graph whose vertices are the graphs with degree sequence \(d\), two vertices are adjacent in the graph \(\mathcal{R}(d)\) if one can be obtained from the other by a switching. It has been shown that the graph \(\mathcal{R}(d)\) is connected. Let \(\mathcal{CR}(d)\) be the set of connected graphs with degree sequence \(d\). Taylor \([13]\) proved that the subgraph of \(\mathcal{R}(d)\) induced by \(\mathcal{CR}(d)\) is connected. Several connected subgraphs of \(\mathcal{CR}(d)(3^n)\) are obtained in this paper. As an application, we are able to obtain the interpolation and extremal results for the number of maximum induced forests in the classes of connected subgraphs of \(\mathcal{CR}(d)(3^n)\).

Let \(T = (V, A)\) be a finite tournament with \(n \geq 2\) vertices. The dual of T is the tournament \(T^* = (V, A^*)\) defined by: for all \(x,y \in V, (x,y) \in A^*\) if and only if \((y,x) \in A\). The tournament \(T\) is critical if \(T\) is indecomposable and if for all \(x \in V\), the subtournament \(T(V – \{x\})\) is decomposable. A \(3\)-cycle is a tournament isomorphic to the tournament \(T, = ({0,1,2}, {(0, 1), (1, 2), (2, 0)})\). Let \(F\) be a set of non negative integers \(k < n\). The tournament \(T\) is \(F\)-selfdual if for every subset \(X\) of \(V\) such that \(|X |\in F\), the subtournaments \(T(X)\) and \(T^*(X)\) are isomorphic. In this paper, we study, for each integer \(k \geq 1\), the \(\{n – k\}\)-selfduality of the tournaments, with \(n \geq 4+k\) vertices, that are lexicographical sums of tournaments under a \(3\)-cycle or a critical tournament. As application, we determine for each integer \(k \geq 1\), the tournaments, with \(n \geq 4+ k\) vertices, that are \(\{4,n – k\}\)-selfdual.

This paper studies in detail the collection of closed sets of a matroid of arbitrary cardinality ordered by inclusion. The relation between the collection, in particular the collection of a simple matroid, and a finite length geometric lattice is dealt with. Finally, one obtains that up to isomorphism, a finite length geometric lattice is a simple matroid, and vice versa.

A vertex set \(D\) of a graph \(G\) is a dominating set if every vertex not in \(D\) is adjacent to some vertex in \(D\). The domination number \(\gamma\) of a graph \(G\) is the minimum cardinality of a dominating set in \(G\).

In 1975, Payan \([6]\) communicated without proof the inequality
\[2\gamma \leq {n} + 1 – \delta\]
for every connected graph not isomorphic to the complement of a one-regular graph, where \(n\) is the order and \(\delta\) the minimum degree of the graph. A first proof of (*) was published by Flach and Volkman \([3]\) in \(1980\).

In this paper, we firstly present a more transparent proof of (*). Using the idea of this proof, we show that

\[2\gamma \leq n – \delta\]

for connected graphs with exception of well-determined families of graphs.