\(n\)-ary hypergroups are a generalization of Dörnte \(n\)-ary groups and a generalization of hypergroups in the sense of Marty. In this paper, we investigate some properties of \(n\)-ary hypergroups and (commutative) fundamental relations. We determine two families \( {P}(H)\) and \( {P}_\sigma(H)\) of subsets of an \(n\)-ary hypergroup \(H\) such that two geometric spaces \((H, {P}(H))\) and \((H, {P}_\sigma(H))\) are strongly transitive. We prove that in every \(n\)-ary hypergroup, the fundamental relation \(\beta\) and the commutative fundamental relation \(\gamma\) are strongly compatible equivalence relations.

In this paper, we develop a technique that allows us to obtain new effective constructions of \(1\)-resilient Boolean functions with very good nonlinearity and autocorrelation. Our strategy to construct a \(1\)-resilient function is based on modifying a bent function by toggling some of its output bits. Two natural questions that arise in this context are: “At least how many bits and which bits in the output of a bent function need to be changed to construct a \(1\)-resilient Boolean function?” We present an algorithm that determines a minimum number of bits of a bent function that need to be changed to construct a \(1\)-resilient Boolean function. We also present a technique to compute points whose output in the bent function need to be modified to get a \(1\)-resilient function. In particular, the technique is applied up to \(14\)-variable functions, and we show that the construction provides \(1\)-resilient functions reaching currently best known nonlinearity and achieving very low autocorrelation absolute indicator values, which were not known earlier.

The noncrossing matchings with each of their blocks containing a given element are introduced and studied. The enumeration of these matchings is described through a polynomial of several variables, which is proved to satisfy a recursive formula. Results of the enumeration of noncrossing matchings with fixed points are connected with Catalan numbers.

For \(1 \leq s \leq n-3\), let \(C_n(i;i_1, v_2, \ldots, i_s)\) denote an \(n\)-cycle with consecutive vertices \(x_1, x_2, \ldots, x_n\) to which the \(s\) chords \(x_{ i}x_{i_1}, x_{i}x_{i_2}, \ldots, x_{i}x_{i_s}\) have been added. In this paper, we discuss the strongly \(c\)-harmonious problem of the graph \(C_n(i;i_1, i_2, \ldots, i_s)\).

A shell of width \(n\) is a fan \(C_n(1;3,4, \ldots, n-1)\) and a vertex with degree \(n-1\) is called apex. \(MS(n^m)\) is a graph consisting of \(m\) copies of shell of width \(n\) having a common apex. If \(m \geq 1\) is odd, then the multiple shell \(MS(n^ m)\) is harmonious.

The closed neighborhood \(N_G[e]\) of an edge \(e\) in a graph \(G\) is the set consisting of \(ev\) and of all edges having a common end-vertex with \(e\). Let \(f\) be a function on \(E(G)\), the edge set of \(G\), into the set \(\{-1, 1\}\). If \(\sum_{x\in E(G)}f(x \geq 1\) for at least \(k\) edges \(e\) of \(G\), then \(f\) is called a signed edge \(k\)-subdominating function of \(G\). The minimum of the values \(\sum_{e \in E(G)} f(e)\), taken over all signed edge \(k\)-subdominating functions \(f\) of \(G\), is called the signed edge \(k\)-subdomination number of \(G\) and is denoted by \(\gamma_{s,k}(G)\). In this note, we initiate the study of the signed edge \(k\)-subdomination in graphs and present some (sharp) bounds for this parameter.

A graph \(G\) with vertex set \(V\) is said to have a prime labeling if its vertices can be labeled with distinct integers \(1, 2, \ldots, |V|\) such that for every edge \(xy\) in \(E(G)\), the labels assigned to \(x\) and \(y\) are relatively prime or coprime. In this paper, we show that the Knödel graph \(W_{3,n}\) is prime for \(n \leq 130\).

Let \(G = (V, E)\) be a graph. A set \(D \subseteq V\) is a total restrained dominating set of \(G\) if every vertex in \(V\) has a neighbor in \(D\) and every vertex in \(V – D\) has a neighbor in \(V – D\). The cardinality of a minimum total restrained dominating set in \(G\) is the total restrained domination number of \(G\). In this paper, we define the concept of total restrained domination edge critical graphs, find a lower bound for the total restrained domination number of graphs, and constructively characterize trees having their total restrained domination numbers achieving the lower bound.

Let \(\Gamma = (X, R)\) denote a \(d\)-bounded distance-regular graph with diameter \(d \geq 3\). A regular strongly closed subgraph of \(\Gamma\) is said to be a subspace of \(\Gamma\). For \(0 \leq i \leq i+s \leq d-1\), suppose \(\Delta_i\) and \(\Delta_0\) are subspaces with diameter \(i\) and \(i+s\), respectively, and with \(\Delta_i \subseteq \Delta_0\). Let \(\mathcal{L}(i, i+s; d)\) denote the set of all subspaces \(\Delta’\) with diameters \(\geq i\) such that \(d(\Delta_0 \cap \Delta’) = \Delta_1\) and \(d(\Delta_0 + \Delta’) = d(\Delta’) + s\) in \(\Gamma$ including \(\Delta_0\). If we partial order \(\mathcal{L}(i, i+s; d)\) by ordinary inclusion (resp. reverse inclusion), then \(\mathcal{L}(i, i+s; d)\) is a poset, denoted by \(\mathcal{L}_0(i, i+s; d)\) (resp. \(\mathcal{L}_R(i, i+s; d)\)). In the present paper, we show that both \(\mathcal{L}_0(i, i+s; d)\) and \(\mathcal{L}_R(i, i+s; d)\) are atomic lattices, and classify their geometricity.

By means of the partial fraction decomposition method, we evaluate a very general determinant of formal shifted factorial fractions, which covers numerous binomial determinantal identities.

Let \(H\) be a subgroup of a finite group \(G\). The relative \(n\)-th commutativity degree, denoted as \(P_n(H,G)\), is the probability of commuting the \(n\)-th power of a random element of \(H\) with an element of \(G\). Obviously, if \(H = G\) then the relative \(n\)-th commutativity degree coincides with the \(n\)-th commutativity degree, \(P_n(G)\). The purpose of this article is to compute the explicit formula for \(P_n(G)\), where \(G\) is a 2-generator \(p\)-group of nilpotency class two. Furthermore, we observe that if we have two pairs of relative isoclinic groups, then they have equal relative \(n\)-th commutativity degree.