Let \(G = (V, E)\) be a simple undirected graph. An independent set is a subset \(S \subseteq V\) such that no two vertices in \(S\) are adjacent. A maximal independent set is an independent set that is not a proper subset of any other independent set.
In this paper, we study the problem of determining the fourth largest number of maximal independent sets among all trees and forests. Extremal graphs achieving these values are also given.

From differential operators and the generating functions of Bernoulli and Euler polynomials, we derive some new theorems on Bernoulli and Euler numbers. By using integral formulae and arithmetical properties relating to the Bernoulli and Euler polynomials, we obtain new identities on Bernoulli and Euler numbers. Finally, we give some new properties on Bernoulli and Euler numbers arising from the \(p\)-adic integrals on \(\mathbb{Z}_p\).

Let \(u,v\) be two vertices of a connected graph \(G\). The vertex \(v\) is said to be a boundary vertex of \(u\) if no neighbor of \(v\) is further away from \(u\) than \(v\). The boundary of a graph is the set of all its boundary vertices.In this work, we present a number of properties of the boundary of a graph under different points of view:(1) A realization theorem involving different types of boundary vertex sets: extreme set, periphery, contour, and the whole boundary.(2) The contour is a monophonic set.(3) The cardinality of the boundary is an upper bound for both the metric dimension and the determining number of a graph.

Computing the crossing number of a given graph is, in general, an elusive problem, and only the crossing numbers of a few families of graphs are known. Most of them are the Cartesian products of special graphs. This paper determines the crossing number of the Cartesian product of a 6-vertex graph with the star \(S_n\).

Let \(M = (E, \mathcal{F})\) be a matroid on a set \(E\), \(B\) one of its bases, and \(M_B\) the base matroid associated to \(B\). In this paper, we determine a characterization of simple binary matroids \(M\) which are not isomorphic to \(M_B\), for every base \(B\) of \(M\). We also extend to matroids some graph notions.

Let \(H\) and \(G\) be two graphs (or digraphs), where \(G\) is a subgraph of \(H\). A \(G\)-decomposition of \(H\), denoted by \((H,G)\)-GD, is a partition of all the edges (or arcs) of \(H\) into subgraphs (\(G\)-blocks), each of which is isomorphic to \(G\). A large set of \((H, G)\)-GD, denoted by \((H, G)\)-LGD, is a partition of all subgraphs isomorphic to \(G\) of \(H\) into \((H,G)\)-GDs. In this paper, we obtain the existence spectra of \((ADK_{m,n}, P_3^i)\)-LGD, where \(P_3^i\) (\(i = 1,2,3\)) are the three types of oriented \(P_3\).

Let \(G\) be a graph. The zeroth-order general Randić index of a graph is defined as \(R_\alpha^0(G) = \sum_{v \in V(G)} d(v)^\alpha(v)\), where \(\alpha\) is an arbitrary real number and \(d(v)\) is the degree of the vertex \(v\) in \(G\). In this paper, we give sharp lower and upper bounds for the zeroth-order general Randić index \(R_\alpha^0(G)\) among all unicycle graphs \(G\) with \(n\) vertices and \(k\) pendant vertices.

\(n\)-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.