In this paper, the congruence relations and the lower and upper bounds of hyper-Wiener index for \(k\)-membered ring spiro systems given length \(n\) are determined respectively. As these results’ applications,the congruence relations and the extremal five- and six-membered ring spiro systems with maximal and minimal hyper-Wiener index are given respectively.

Let \(G\) be a finite group and \(S \subseteq G \setminus \{0\}\). We call \(S\) an additive basis of \(G\) if every element of \(G\) can be expressed as a sum over a nonempty subset in some order. Let \(cr(G)\) be the smallest integer \(t\) such that every subset of \(G \setminus \{0\}\) of cardinality \(t\) is an additive basis of \(G\). In this paper, we determine \(cr(G)\) for the following cases: (i) \(G\) is a finite nilpotent group; (ii) \(G\) is a group of even order which possesses a subgroup of index \(2\).

For \(n \geq 1\), we let \(a_n\) count the number of compositions of the positive integer \(m\), where the last summand is odd. We find that \(a_n = (\frac{1}{3})(-1)^n + (\frac{2}{3}) 2^{n-1}\). Since \(J_n\), the \(n\)-th Jacobsthal number, is given as \(\frac{1}{3}(-1)^n + \frac{2}{3}2^{n-1}\) for \(n \geq 0\), it follows that \(a_n = J_{n-1}\) for \(n \geq 1\). For this reason, these compositions are often referred to as the Jacobsthal compositions.

In our investigation, we determine results for the \(a_n\) compositions of \(n\), such as: (i) \(a_{n,k}\), the number of times the positive integer \(k\) appears as a summand among these \(a_n\) compositions of \(n\); (ii) the numbers of plus signs, summands, even summands, and odd summands that occur for these compositions of \(n\); (iii) the sum of the even summands and the sum of the odd summands for the \(a_n\) compositions of \(n\); (iv) the numbers of levels, rises, and descents for the \(a_n\) compositions; and (v) the number of runs that occur among these \(a_n\) compositions.

In this paper, we introduce a new sequence called standard Young words, which are defined as quaternary words with interesting restrictions. First, we show that the cardinality of standard Young words of length n is related to Catalan triangle sequence and we establish a bijection from the set of standard Young words to the set of pairs of non-intersection lattice paths. Then we set a one-to-one correspondence between the set of standard Young words and the set of standard Young tableaux of two rows, which results in the correspondence between the statistics of standard Young words and standard Young tableaux, such as sign and descents.

A graph \(G\) is called a fractional \((k, m)\)-deleted graph if after deleting any \(m\) edges of \(G\), the resulting graph admits a fractional \(k\)-factor. In this paper, we prove that for \(k \geq 2\) and \(m \geq 0\), \(G\) is a fractional \((k, m)\)-deleted graph if one of the following conditions holds: 1) \(n \geq 4k + 4m – 3\), \(\delta(G) \geq k + m\), and \(\max\{d_G(u), d_G(v)\} \geq \frac{n}{2}\) for each pair of non-adjacent vertices \(u\) and \(v\) of \(G\); 2) \(\delta(G) \geq k + m\), \(\omega_2(G) \geq n\), \(n \geq 4k + 4m – 5\) if \((k, m) = (3, 0)\), and \(n \geq 8\) if \((k, m) = (3, 0)\). The results are best possible in some sense.

Let \(K\) be a real quadratic field \(\mathbb{Q}(\sqrt{n})\) with an integer \(n = df^2\), where \(d\) is the field discriminant of \(K\) and \(f \geq 1\). Q. Mushtaq found an interesting phenomenon that any totally negative number \(\kappa_0\) with \(\kappa^{\sigma} < 0\) and \(\kappa_0^{\sigma} < 0\) belonging to the discriminant \(n\), attains an ambiguous number \(\kappa_m\) with \(\kappa_m \kappa_m^{\sigma} < 0\) after finitely many actions \(\kappa_0^{A_j}\) with \(0 \leqq j \leqq m\) by modular transformations \(A_j \in \mathrm{SL}_2^+(\mathbb{Z})\). Here \(\sigma\) denotes the embedding of \(K\) distinct from the identity. In this paper, we give a new aspect for the process to reach an ambiguous number from a totally negative or totally positive number, by which the gap of the proof of Q. Mushtaq's Theorem is complemented. Next, as an analogue of Gauss' Genus Theory, we prove that the ring class number \(h_{+}(df^2)\) coincides with the ambiguous class number belonging to the discriminant \(n = df^2\), and its behavior is unbounded when \(f\) with suitable prime factors goes to infinity using the ring class number formula.

For a rational number \(r > 1\), a set \(A\) of positive integers is called an \(r\)-multiple-free set if \(A\) does not contain any solution of the equation \(rx = y\). The extremal problem of estimating the maximum possible size of \(r\)-multiple-free sets contained in \([n] := \{1, 2, \ldots, n\}\) has been studied in combinatorial number theory for theoretical interest and its application to coding theory. Let \(a\) and \(b\) be relatively prime positive integers such that \(a < b\). Wakeham and Wood showed that the maximum size of \((b/a)\)-multiple-free sets contained in \([n]\) is \( \frac{b}{b+1} + O(\log n)\). In this note, we generalize this result as follows. For a real number \(p \in (0, 1)\), let \([n]_p\) be a set of integers obtained by choosing each element \(i \in [n]\) randomly and independently with probability \(p\). We show that the maximum possible size of \((b/a)\)-multiple-free sets contained in \([n]_p\) is \({\frac{b}{b+p}pn} + O(\sqrt{pn} \log n \log \log n)\) with probability that goes to \(1\) as \(n \to \infty\).

A partition of an integer \(n\) is a representation \(n = a_1 + a_2 + \cdots + a_k\), with integer parts \(a_1 \geq a_2 \geq \cdots \geq a_k \geq 1\). The Durfee square is the largest square of points in the graphical representation of a partition. We consider generating functions for the sum of areas of the Durfee squares for various different classes of partitions of \(n\). As a consequence, interesting partition identities are derived. The more general case of Durfee rectangles is also treated, as well as the asymptotic growth of the mean area over all partitions of \(n\).

A graph \(G\) is called a fractional \((k, m)\)-deleted graph if any \(m\) edges are removed from \(G\), then the resulting graph admits a fractional \(k\)-factor. In this paper, we prove that for integers \(k \geq 2\), \(m \geq 0\), \(n \geq 8k + 4m – 7\), and \(\delta(G) \geq k + m\), if
\[|N_G(x) \cup N_G(y)| \geq \frac{n}{2}\]
for each pair of non-adjacent vertices \(x, y\) of \(G\), then \(G\) is a fractional \((k, m)\)-deleted graph. The bounds for neighborhood union condition, order, and the minimum degree of \(G\) are all sharp.

A \(c\)-partite or multipartite tournament is an orientation of a complete \(c\)-partite graph. A digraph \(D\) is cycle complementary if there exist two vertex-disjoint directed cycles \(C\) and \(C’\) such that \(V(D) = V(C) \cup V(C’)\). The global irregularity of a digraph \(D\) is defined by
\[i_g(D) = \max\{\max(d^+(x), d^-(x)) – \min(d^+(y),d^-(y)) \mid x,y \in V(D)\}.\]
If \(i_g(D) = 0\), then \(D\) is regular, and if \(i_g(D) \leq 1\), then \(D\) is almost regular. We prove in this paper that every almost regular \(c\)-partite tournament with \(c \geq 3\) such that all partite sets have the same cardinality \(r \geq 4\) contains two complementary directed cycles of length \(3\) and \(|V(D)| – 3\).

In this paper, we determine the spectrum for \(super-perfect\) OQSs. OQSs are \(G\)-designs in which \(G\) is an octagon quadrangle, i.e., the graph consisting of an \(8\)-cycle \((x_1, x_2, \ldots, x_8)\) with two additional chords: the edges \(\{x_1, x_4\}\) and \(\{x_5, x_6\}\).

In this paper, we give a four parameter theta function identity and prove it by using some properties of Jacobi’s theta functions and Jacobi’s fundamental formulae.

The order dimension is an invariant on partially ordered sets introduced by Dushnik and Miller in \(1941 [1]\). It is known that the computation of the order dimension of a partially ordered set in general is highly complex,with current algorithms relying on the minimal coloring of an associated hypergraph, see \([5]\). The aim of this work is to extend the family of posets whose order dimension is easily determined by a formula. We introduce an operation called layering. Finally, we provide the precise formulas for determining the order dimension of any given number of layers of Trotter’s generalized crowns.

In this paper, the regular endomorphisms of a split graph are investigated. We give a condition under which the regular endomorphisms of a split graph form a monoid.

The clique graph \(K(G)\) of a graph \(G\) is the intersection graph of all its (maximal) cliques, and \(G\) is said to be clique divergent if the order of its \(n\)-th iterated clique graph \(K^n(G)\) tends to infinity with \(n\). In general, deciding whether a graph is clique divergent is not known to be computable. We characterize the dynamical behavior under the clique operator of circulant graphs of the form \(C_n(a, b, c)\) with \(0 < a < b < c < \frac{n}{3}\). Such a circulant is clique divergent if and only if it is not clique-Helly. Owing to the Dragan-Szwarcfiter Criterion to decide clique-Hellyness, our result implies that the clique divergence of these circulants can be decided in polynomial time. Our main difficulty was the case \(C_n(1, 2, 4)\), which is clique divergent but no previously known technique could be used to prove it.

A total dominating set \(S\) of a graph \(G\) with no isolated vertex is a locating-total dominating set of \(G\) if for every pair of distinct vertices \(u\) and \(v\) in \(V – S\) are totally dominated by distinct subsets of the total dominating set. The minimum cardinality of a locating-total dominating set is the locating-total domination number. In this paper, we obtain new upper bounds for locating-total domination numbers of the Cartesian product of cycles \(C_m\) and \(C_n\), and prove that for any positive integer \(n \geq 3\), the locating-total domination numbers of the Cartesian product of cycles \(C_3\) and \(C_n\) is equal to \(n\) for \(n \equiv 0 \pmod{6}\) or \(n + 1\) otherwise.

A graph \(G\) is called a fractional \((g, f, m)\)-deleted graph if after deleting any \(m\) edges, then the resulting graph admits a fractional \((g, f)\)-factor. In this paper, we prove that if \(G\) is a graph of order \(n\), and if \(1 \leq g(x) \leq f(x) \leq 6\) for any \(x \in V(G)\), \(\delta(G) \geq \frac{b^2(i-1)}{a} ++2m\), \(n > \frac{(a+b)(i(a+b)+2m-2)}{a}\) and \(|N_G(x_1) \cup N_G(x_2) \cup \cdots \cup N_G(x_i)| \geq \frac{bn}{a+b} \), for any independent set \(\{x_1, x_2, \ldots,x_i\}\) of \(V(G)\), where \(i \geq 2\), then \(G\) is a fractional \((g, f, m)\)-deleted graph. The result is tight on the neighborhood union condition.

In this short paper, we introduce the second order linear recurrence relation of the \(AB\)-generalized Fibonacci sequence and give the explicit formulas for the sums of the positively and negatively subscripted terms of the \(AB\)-generalized Fibonacci sequence by matrix methods. This sum generalizes the one obtained earlier by Kilig in \([2]\).

Only few results concerning crossing numbers of join of some graphs are known. In the paper, for the special graph \(G\) on six vertices, we give the crossing numbers of \(G\vee P_n\) and \(G\vee C_n\), \(P_n\) and \(C_n\) are the path and cycle on \(n\) vertices, respectively.

Recently, Dere and Simsek have treated some applications of umbral algebra. related to several special polynomials(see \([8]\)). In this paper, we derive some new and interesting identities of special polynomials involving Bernoulli, Euler and Laguerre polynomials arising from umbral calculus.

In this paper, we prove that for any tree \(T\), \(T^2\) is a divisor graph if and only if \(T\) is a caterpillar and the diameter of \(T\) is less than six. For any caterpillar \(T\) and a positive integer \(k \geq 1\) with \(diam(T) \leq 2k\), we show that \(T^k\) is a divisor graph. Moreover, for a caterpillar \(T\) and \(k \geq 3\) with \(diam(T) = 2k\) or \(diam(T) = 2k + 1\), we show that \(T^k\) is a divisor graph if and only if the centers of \(T\) have degree two.

To construct a large graph from two smaller ones that have same order, one can add an arbitrary perfect matching between their vertex-sets. The topologies of many networks are special cases of these graphs. An interesting and important problem is how to persist or even improve their link reliability and link fault-tolerance. Traditionally, this may be done by optimizing the edge connectivity of their topologies, a more accurate method is to improve their \(m\)-restricted edge connectivity. This work presents schemes for optimizing \(m\)- restricted edge connectivity of these graphs, some well-known results are direct consequences of our observations.

In this paper we introduce a new kind of generalized Pell numbers. This generalization is introduced in the distance sense. We give different interpretations and representations of these numbers.We present relations between distance Pell numbers and Fibonacci numbers. Moreover we describe graph interpretations of distance Pell numbers. These graphs interpretations in the natural way imply a new kind of generalized Jacobsthal numbers.

A graph \(G\) is called a fractional \((g, f, n’, m)\)-critical deleted graph if after deleting any \(n’\) vertices of \(G\) the remaining graph is a fractional \((g, f, m)\)-deleted graph. In this paper, we give two binding number conditions for a graph to be a fractional \((g, f, n’, m)\)-critical deleted graph.

In this paper, we compute the hyper-Wiener index of arbitrary \(k\)-membered ring spiro chain. We also determine the extremal \(k\)-membered ring spiro chains for hyper-Wiener index.

In this paper, the notion of cyclic bursts in array codes equipped with a non-Hamming metric \([13]\) as a generalization of classical cyclic bursts \([5]\) is introduced and some bounds are obtained on the parameters of array codes for the detection and correction of cyclic burst array errors.

Let \(G\) be a graph, and let \(a\), \(b\), \(k\) be integers with \(0 \leq a \leq b\), \(k \geq 0\). An \([a, b]\)-factor of graph \(G\) is defined as a spanning subgraph \(F\) of \(G\) such that \(a \leq d_F(v) \leq b\) for each \(v \in V(F)\). Then a graph \(G\) is called an \((a, b, k)\)-critical graph if after deleting any \(k\) vertices of \(G\) the remaining graph of \(G\) has an \([a, b]\)-factor. In this paper, it is proved that, if \(a\), \(b\), \(k\) be integers with \(1 \leq a < b\), \(k \geq 0\) and \(b \geq a(k+1)\) and \(G\) is a graph with \(\delta(G) \geq a+k\) and binding number \(b(G) \geq a-1+\frac{a(k+1)}{b}\), then \(G\) is an \((a, b, k)\)-critical graph. Furthermore, it is shown that the result in this paper is best possible in some sense.