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.

In this note, we consider one type of \(k\)-tridiagonal matrix family whose permanents and determinants are specified to the balancing and Lucas-balancing numbers. Moreover, we provide some properties between Chebyshev polynomial properties and the given number
sequences,

Let \(G = (V, E)\) be a graph. A subset \(D \subseteq V\) is a dominating set if every vertex not in \(D\) is adjacent to a vertex in \(D\). The domination number of \(G\) is the smallest cardinality of a dominating set of \(G\). The bondage number of a nonempty graph \(G\) is the smallest number of edges whose removal from \(G\) results in a graph with larger domination number than \(G\). In this paper, we determine that the exact value of the bondage number of an \((n-3)\)-regular graph \(G\) of order \(n\) is \(n-3\).

A graph is closed when its vertices have a labeling by \([n]\) with a certain property first discovered in the study of binomial edge ideals. In this article, we prove that a connected graph has a closed labeling if and only if it is chordal, claw-free, and has a property we call narrow, which holds when every vertex is distance at most one from all longest shortest paths of the graph.

Let \(\Sigma = (X, \mathcal{B})\) be a \(4\)-cycle system of order \(v = 1 + 8k\). A \(c\)-colouring of type \(s\) is a map \(\phi: \mathcal{B} \to C\), where \(C\) is a set of colours, such that exactly \(c\) colours are used and for every vertex \(x\), all the blocks containing \(x\) are coloured exactly with \(s\) colours. Let \(4k = qs + r\), with \(r \geq 0\). \(\phi\) is equitable if for every vertex \(x\), the set of the \(4k\) blocks containing \(x\) is partitioned into \(r\) colour classes of cardinality \(q + 1\) and \(s – r\) colour classes of cardinality \(q\). In this paper, we study colourings for which \(s | k\), providing a description of equitable block colourings for \(c \in \{s, s+1, \ldots, \lfloor \frac{2s^2+s}{3} \rfloor\}\).