A three-colored digraph \(D\) is primitive if and only if there exist nonnegative integers \(h\), \(k\), and \(v\) with \(h+k+v > 0\) such that for each pair \((i, j)\) of vertices there is an \((h, k, v)\)-walk in \(D\) from \(i\) to \(j\). The exponent of the primitive three-colored digraph \(D\) is defined to be the smallest value of \(h + k + v\) over all such \(h\), \(k\), and \(v\). In this paper, a class of special primitive three-colored digraphs with \(n\) vertices, consisting of one \(n\)-cycle and two \((n-1)\)-cycles, are considered. For the case \(a = c – 1\), some primitive conditions, the tight upper bound on the exponents, and the characterization of extremal three-colored digraphs are given.

Skew-quasi-cyclic codes over a finite field are viewed as skew-cyclic codes on a noncommutative ring of matrices over a finite field. This point of view gives a new construction of skew-quasi-cyclic codes. Let \(\mathbb{F}_q\) be the Galois field with \(q\) elements and \(\theta\) be an automorphism of \(\mathbb{F}_q\). We propose an approach to consider the relationship between left ideals in \(M_l(\mathbb{F}_q)[X, \theta]/(X^s – 1)\) and skew-quasi-cyclic codes of length \(ls\) and index \(l\) over \(\mathbb{F}_q\), under \(\theta\), which we denote by \(\theta\)-SQC codes (or SQC codes for short when there is no ambiguity). We introduce the construction of SQC codes from the reversible divisors of \(X^s – 1\) in \(M_l(\mathbb{F}_q)[X, \theta]\). In addition, we give an algorithm to search for the generator polynomials of general SQC codes.

In this paper, we investigate the concepts of \(k\)-limited packing and \(k\)-tuple domination in graphs and give several bounds on the size of them. These bounds involve many well-known parameters of graphs. Also, we establish a connection between these concepts that implies some new results in this area. Finally, we improve many bounds in the literature.

Let \(G = (V, E)\) be a simple graph. A paired-dominating set of a graph \(G\) is a dominating set whose induced subgraph contains a perfect matching. The paired domination number of a graph \(G\), denoted by \(\gamma_p(G)\), is the minimum cardinality of a paired-dominating set in \(G\). In this paper, we study the paired domination number of generalized Petersen graphs \(P(n,2)\) and prove that for any integer \(n \geq 6\), \(\gamma_p(P(n, 2)) = 2 \left\lfloor \frac{n}{3} \right\rfloor + n \pmod{3}\).

The Estrada index of a simple connected graph \(G\) of order \(n\) is defined as \(EE(G) = \sum_{i=1}^{n} e^{\lambda_i}\), where \(\lambda_1, \lambda_2, \ldots, \lambda_n\) are the eigenvalues of the adjacency matrix of \(G\). In this paper, we characterize all pentacyclic graphs of order \(n\) with maximal Estrada index.

Let \(\Pi\) be a finite polar space of rank \(n \geq 2\) fully embedded into a projective space \(\Sigma\). In this note, we determine all tight sets of \(\Pi\) of the form \((\Sigma_1 \cap \mathcal{P}) \cup (\Sigma_2 \cap \mathcal{P})\), where \(\mathcal{P}\) denotes the point set of \(\Pi\) and \(\Sigma_1, \Sigma_2\) are two mutually disjoint subspaces of \(\Sigma\). In this way, we find two families of \(2\)-tight sets of elliptic polar spaces that were not described before in the literature.

In this paper, we define a new matrix identity for bi-periodic Fibonacci and Lucas numbers. By using the matrix method, we give simple proofs of several properties of these numbers. Moreover, we obtain a new binomial sum formula for bi-periodic Fibonacci and Lucas numbers, which generalize the former results.

Hein and Sarvate show how to decompose \(\lambda\) copies of a complete graph \(K_n\), for some minimal value of \(\lambda\), into so-called LOE and OLE graphs. In this paper, we will show that for all possible values of \(\lambda\), the necessary conditions are sufficient for the LOE and OLE decompositions.

Let \(R\) be a commutative ring. The regular digraph of ideals of \(R\), denoted by \(\mathcal{R}(R)\), is a digraph whose vertex-set is the set of all non-trivial ideals of \(R\) and, for every two distinct vertices \(I\) and \(J\), there is an arc from \(I\) to \(J\), whenever \(I\) contains a non-zero divisor of \(J\). In this paper, we investigate the planarity of \(\mathcal{R}(R)\). We also completely characterize the rings \(R\) such that \(\mathcal{R}(R)\) is a ring graph, and the situations under which the genus of \(\mathcal{R}(R)\) is finite. Moreover, we study the independence number and the girth of \(\mathcal{R}(R)\), and also we find all cases that \(\mathcal{R}(R)\) is bipartite.

In this paper, the existence of Yang Hui type magic squares of order \(n\) with \(t\)-powered sum (YMS(\(n\), \(t\))) for general \(t\) is investigated. Some constructions of YMS(\(n\), \(t\)) are obtained by using strongly symmetric self-orthogonal diagonal Latin squares and magic rectangles. Applying these constructions, it is proved that for an integer \(t > 1\) there exist both a symmetric elementary YMS(\(2^t\), \(2t – 2\)) and a symmetric elementary YMS(\(2^t – k\), \(2t\)) for odd \(k > 1\), which improves the known result on YMSs.