The commuting graph of an arbitrary ring \(R\), denoted by \(\Gamma(R)\), is the graph whose vertices are all non-central elements of \(R\), and two distinct vertices \(a\) and \(b\) are adjacent if and only if \(ab = ba\). In this paper, we investigate the connectivity, the diameter, the maximum degree and the minimum degree of the commuting graph of the quaternion algebra \(\mathbb{Z}_n[i, j, k]\).

A set \(D\) of vertices of a graph \(G = (V, E)\) is a \(\textit{dominating set}\) if every vertex of \(V-D\) is adjacent to at least one vertex in \(D\). The \(\textit{domination number}\) \(\gamma(G)\) is the minimum cardinality of a dominating set of \(G\). A subset of \(V-D\), which is also a dominating set of \(G\), is called an \(\textit{averse dominating set}\) of \(G\) with respect to \(D\). The \(\textit{inverse domination number}\) \(\gamma'(G)\) equals the minimum cardinality of an inverse dominating set \(D\). In this paper, we study classes of graphs whose domination and inverse domination numbers are equal.

A strongly connected digraph \(\Gamma\) is said to be walk regular if for any nonnegative integer \(l\) and any vertex \(u\) of \(\Gamma\), the number of circuits of length \(l\) containing \(u\) depends only on \(l\). This family of digraphs is a directed version of walk regular graphs. In this paper, we discuss some basic properties of walk regular digraphs.

For any \(n \geq 2\) we let \(S_n\) be the set of permutations of the set \(\{1,2,\ldots,n\}\). A reduction \(\overline{f}\) on \(S_n\) is a set of functions \(\{f_i : 1 \leq i \leq n\}\) such that \(f_n\) is the identity function on \(\{1,2,\ldots,n-1\}\) and for \(i n_0\), such that \(\phi(n) \leq n\) for all \(n \geq n_0\), and for which \(p \downarrow \phi(n) \downarrow i = p \downarrow i \downarrow n-1\) for all \(n > n_0\), for all \(i \leq n-1\) and for all \(p \in S_n\). And the system is said to be amenable if for every \(n > n_0\) there is an integer \(k < n\) such that, for all \(p \in S_n\), \(p \downarrow k \downarrow n-1 = p \downarrow n-1\). The purpose of this paper is to study faithful reductions and linked reduction systems. We characterize amenable, linked reduction systems by means of two types of liftings by which a reduction on \(S_{n+1}\) can be formed from one on \(S_n\). And we obtain conditions for a reduction system to be faithful. One interesting consequence is that any amenable, linked reduction system which begins with a simple reduction is faithful.

Let \(N\) be a positive integer and let \(\lambda = (\lambda_1, \lambda_2, \ldots, \lambda_l)\) be a partition of \(N\) of length \(l\), i.e., \(\sum_{i=1}^{l}\lambda_i = N\) with parts \(\lambda_1 \geq \lambda_2 \geq \cdots \geq \lambda_l \geq 1\). Define \(T(\lambda)\) as the partition of \(N\) with parts \(l\), \(\lambda_1 – 1, \lambda_2 – 1, \ldots, \lambda_l – 1\), ignoring any zeros that might occur. Starting with a partition \(\lambda\) of \(N\), we describe Bulgarian Solitaire by repeatedly applying the shift operation \(T\) to obtain the sequence of partitions

\[\lambda, T(\lambda), T^2(\lambda), \ldots\]

We say a partition \(\mathcal{A}\) of \(N\) is \(T\)-cyclic if \(T^i(\mu) = \mu\) for some \(i \geq 1\). Brandt \([2]\) characterized all \(T\)-cyclic partitions for Bulgarian Solitaire. In this paper, we give an inductive proof of Brandt’s result.

Let \(D\) be an acyclic digraph. The competition graph of \(D\) is a graph which has the same vertex set as \(D\) and has an edge between \(x\) and \(y\) if and only if there exists a vertex \(v\) in \(D\) such that \((x, v)\) and \((y, v)\) are arcs of \(D\). For any graph \(G\), \(G\) together with sufficiently many isolated vertices is the competition graph of some acyclic digraph. The competition number \(k(G)\) of \(G\) is the smallest number of such isolated vertices.
A hole of a graph is a cycle of length at least \(4\) as an induced subgraph. In \(2005\), Kim \([5]\) conjectured that the competition number of a graph with \(h\) holes is at most \(h + 1\). Though Li and Chang \([8]\) and Kim et al. \([7]\) showed that her conjecture is true when the holes do not overlap much, it still remains open for the case where the holes share edges in an arbitrary way. In order to share an edge, a graph must have at least two holes and so it is natural to start with a graph with exactly two holes. In this paper, the conjecture is proved true for such a graph.

Let \(G\) be a graph with domination number \(\gamma(G)\). A dominating set \(S \subseteq V(G)\) has property \(\mathcal{UK}\) if all components of the subgraph it induces in \(G\) are complete. The union of complete graphs domination number of a graph \(G\), denoted \(\gamma_{uk}(G)\), is the minimum possible size of a dominating set of \(G\), which has property \(\mathcal{UK}\). Results on changing and unchanging of \(\gamma_{uk}(G)\) after vertex removal are presented. Also forbidden subgraph conditions sufficient to imply \(\gamma(G) = \gamma_{uk}(G)\) are given.

In this paper, we use the finite Heine \({}_{2}\Phi_1\) transformations given in \([4]\) and some elementary simplifications to obtain several Rogers-Ramanujan type identities.

Using lines in a two-dimensional vector space \({GF}(q^2)\) over \({GF}(q)\), we construct some classes of external difference families over \({GF}(q^2)\).