We generalize the well known congruence Lucas\(^1\) Theorem for binomial coefficient to the bi\(^s\)nomial coefficients.

The linear arboricity \(la(G)\) of a graph \(G\) is the minimum number of linear forests that partition the edges of \(G\). In this paper, it is proved that if \(G\) is a planar graph with maximum degree \(\Delta \geq 7\) and every \(7\)-cycle of \(G\) contains at most two chords, then \(la(G) = \left\lceil \frac{\Delta(G)}{2} \right\rceil\).

In this paper, we study the generalized Pell \(p\)-sequences modulo \(m\). Additionally, we define the generalized Pell \(p\)-sequences and the basic generalized Pell sequences in groups, and then examine these sequences in finite groups. Furthermore, we obtain the periods of the generalized Pell \(p\)-sequences and the basic periods of the basic generalized Pell sequences in the binary polyhedral groups \(\langle n,2,2\rangle\), \(\langle2,n,2\rangle\), and \(\langle2,2,n\rangle\).

The matching preclusion number of a graph is the minimum number of edges whose deletion results in a graph that has neither perfect matchings nor almost-perfect matchings. For many interconnection networks, the optimal sets are precisely those incident to a single vertex. Recently, the conditional matching preclusion number of a graph was introduced to look for obstruction sets beyond those incident to a single vertex. It is defined as the minimum number of edges whose deletion results in a graph with no isolated vertices that has neither perfect matchings nor almost-perfect matchings. In this paper, we find this number and classify all optimal sets for the star graphs, one of the most popular interconnection networks.

The terminal Wiener index of a tree is the sum of distances for all pairs of pendent vertices, which recently arose in the study of phylogenetic tree reconstruction and the neighborhood of trees. This paper presents sharp upper and lower bounds for the terminal Wiener index in terms of its order and diameter and characterizes all extremal trees that attain these bounds. Additionally, we investigate the properties of extremal trees that attain the maximum terminal Wiener index among all trees of order \(n\) with fixed maximum degree.

Based on some results of Shult and Yanushka [7], Brouwer [1] proved that there exists a unique regular near hexagon with parameters \((s,t,t_2) = (2,11,1)\), namely the one related to the extended ternary Golay code. His proof relies on the uniqueness of the Witt design \(S(5,6,12)\), Pless’s characterization of the extended ternary Golay code \(G_{12}\), and some properties of \(S(5,6,12)\) and \(G_{12}\). It is possible to avoid all this machinery and provide an alternative, more elementary and self-contained proof for the uniqueness. The author recently observed that such an alternative proof is implicit in the literature, obtainable by combining results from [1], [4], and [7]. This survey paper aims to bring this fact to the attention of the mathematical community. We describe the relevant parts of the above papers for this alternative proof of classification. Additionally, we prove several extra facts not explicitly contained in [1], [4], or [7]. This paper can also be seen as an addendum to Section 6.5 of [3], where the uniqueness of the near hexagon was not proved.

Recently, Belbachir and Belkhir gave some recurrence relations for the \(r\)-Lah numbers. In this paper, we give other properties for the \(r\)-Lah numbers, we introduce and study a restricted class of these numbers.

An \(H\)-polygon is a simple polygon whose vertices are \(H\)-points, which are points of the set of vertices of a tiling of \(\mathbb{R}^2\) by regular hexagons of unit edge. Let \(G(v)\) denote the least possible number of \(H\)-points in the interior of a convex \(H\)-polygon \(K\) with \(v\) vertices. In this paper, we prove that \(G(8) = 2\), \(G(9) = 4\), \(G(10) = 6\), and \(G(v) \geq \lceil \frac{v^2}{16\pi^2}-\frac{v}{4}+\frac{1}{2}\rceil – 1\) for all \(v \geq 11\), where \(\lceil x \rceil\) denotes the minimal integer more than or equal to \(x\).

Row-cyclic array codes have already been introduced by the author \([9]\). In this paper, we give some special classes of row-cyclic array codes as an extension of classical BCH and Reed-Solomon codes.

The harmonic weight of an edge is defined as reciprocal of the average degree of its end-vertices. The harmonic index of a graph \(G\) is defined as the sum of all harmonic weights of its edges. In this work, we give the minimum value of the harmonic index for any \(n\)-vertex connected graphs with minimum degree \(\delta\) at least \(k(\geq n/2)\) and show the corresponding extremal graphs have only two degrees,i.e., degree \(k\)and degree \(n – 1\), and the number of vertices of degree \(k\) is as close to \(n/2\) as possible.