The main purpose of this paper is to define a pair of Konhauser matrix polynomials and obtain some properties, such as recurrence relations and matrix differential equations, for Konhauser matrix polynomials.

Studying expressions of the form \((f(z)D)^n\), where \(D = \frac{d}{dx}\) is the derivation operator, goes back to Scherk’s Ph.D. thesis in 1823. We show that this can be extended as
\(\sum{\gamma_{p;a}}(f^{(0)})^{a(0)+1}(f^{(1)})^{a(1)}\ldots (f^{(p-1)})^{a(p-1)}D^{p-\sum_i ia(i)},\) where the summation is taken over the \(p\)-tuples \((a_0, a_1, \ldots, a_{p-1})\), satisfying \(\sum_ia(i)=p-1 + ,\sum_iia(i) < p\), \(f^{(i)} = D^if\), and \(\gamma_{p;a}\) is the number of increasing trees on the vertex set \([0, p]\) having \(a(0) + 1\) leaves and having \(a(i)\) vertices with \(i\) children for \(0 < i < p\). Thus, previously known results about increasing trees lead us to some equalities containing coefficients \(\gamma_{p;a}\). In the sequel, we consider the expansion of \({(x^kD)}^p\) and coefficients appearing there, which are called generalized Stirling numbers by physicists. Some results about these coefficients and their inverses are discussed through bijective methods. Particularly, we introduce and use the notion of \((p,k)\)-forest in these arguments.

In this note, we determine the exact value for the second largest eigenvalue of the derangement graph, by deriving a formula for all the eigenvalues corresponding to the \(2\)-part partitions. This result is then used to obtain.

Since ancient times, mathematicians have considered geometrical objects with integral side lengths. We consider plane integral point sets \(P\), which are sets of \(n\) points in the plane with pairwise integral distances, where not all the points are collinear.

The largest occurring distance is called its diameter. Naturally, the question about the minimum possible diameter \(d(2, 7)\) of a plane integral point set consisting of \(7\) points arises. We give some new exact values and describe state-of-the-art algorithms to obtain them. It turns out that plane integral point sets with minimum diameter consist very likely of subsets with many collinear points. For this special kind of point sets, we prove a lower bound for \(d(2, n)\) achieving the known upper bound \(n^{c_2\log \log n }\) up to a constant in the exponent.
A famous question of Erdés asks for plane integral point sets with no \(3\) points on a line and no \(4\) points on a circle. Here, we talk of point sets in general position and denote the corresponding minimum diameter by \(d(2,n)\). Recently \(d(2, 7) = 22270\) could be determined via an exhaustive search.

In this paper, we study invariant sequences by umbral method, and give some identities which are similar with the identities of Bernoulli numbers.

In this paper, we consider the total domination number, the restrained domination number, the total restrained domination number and the connected domination number of lexicographic product graphs.

In this paper, we obtain the numbers of embeddings of wheel graphs on some orientable and nonorientable surfaces of small genera, mainly on torus, double torus, and nonorientable surfaces of genus \(1, 2, 3\), and \(4\). These are the first results for embeddings of wheel graphs on nonorientable surfaces as known up to now.

An \((a, d)\)-edge-antimagic total labeling for a graph \(G(V, E)\) is an injective mapping \(f\) from \(V \cup E\) onto the set \(\{1, 2, \ldots, |V| + |E|\}\) such that the set \(\{f(v) + \sum f(uv) \mid uv \in E\}\), where \(v\) ranges over all of \(V\), is \(\{a, a+d, a+2d, \ldots, a+(|V|-1)d\}\). Simanjuntak et al conjecture:1. \(C_{2n}\) has a \((2n + 3, 4)\)- or a \((2n + 4, 4)\)-edge-antimagic total labeling;
2. cycles have no \((a, d)\)-edge-antimagic total labelings with \(d > 5\).In this paper, these conjectures are shown to be true.

This article discusses the geometricity of the direct sum, direct product and lexicographic products of two lattices, and compute their characteristic polynomials and classify their geometricity.

This paper introduces the concepts of a \({supergraph}\) and \({graphical\; complexity}\) of a permutation group, intended as a tool for investigating the structure of concrete permutation groups. Basic results are established and some research problems suggested.