In this article, the lines not meeting a hyperbolic quadric in PG\((3,q)\) are characterized by their intersection properties with points and planes.

By the classical method for obtaining the values of the Riemann zeta-function at even positive integral arguments, we shall give some functional equational proof of some interesting identities and recurrence relations related to the generalized higher-order Euler and Bernoulli numbers attached to a Dirichlet character \(\chi\) with odd conductor \(d\), and shall show an identity between generalized Euler numbers and generalized Bernoulli numbers. Finally, we remark that any weighted short-interval character sums can be expressed as a linear combination of Dirichlet \(L\)-function values at positive integral arguments, via generalized Bernoulli (or Euler) numbers.

A point set \(X\) in the plane is called a k-distance set if there are exactly \(k\) different distances between two distinct points in \(X\). We classify \(11\)-point \(5\)-distance sets.

In this paper, we define the self-inverse sequences related to Sheffer sets and give some interesting results of these sequences. Moreover, we study the self-inverse sequences related to the Laguerre polynomials of order \(a\).

Assume we have a set of \(k\) colors and we assign an arbitrary subset of these colors to each vertex of a graph \(G\). If we require that each vertex to which an empty set is assigned has in its neighborhood all \(k\) colors, then this assignment is called the \(k\)-rainbow dominating function of a graph \(G\). The minimum sum of numbers of assigned colors over all vertices of \(G\), denoted as \(\gamma_{rk}(G)\), is called the \(k\)-rainbow domination number of \(G\). In this paper, we prove that \(\gamma_{r2}(P(n, 3)) \geq \left\lceil \frac{7n}{8} \right\rceil.\)

Let \(G\) be a graph with vertex set \(V(G)\), and let \(k \geq 2\) be an integer. A spanning subgraph \(F\) of \(G\) is called a fractional \(k\)-factor if \(d_G^h(x) = k\) for all \(x \in V(G)\), where \(d_G^h(x) = \sum_{e \in E_x} h(e)\) is the fractional degree of \(x \in V(F)\) with \(E_x = \{e : e = xy, e \in E(G)\}\). The binding number \(bind(G)\) is defined as follows:

\[bind(G) = \min\left\{\frac{|N_G(X)|}{|X|} :\varnothing \neq X \subseteq V(G), N_G(G) \neq V(G)\right\}.\]

In this paper, a binding number condition for a graph to have fractional \(k\)-factors is given.

Let \(\Gamma\) denote a \(d\)-bounded distance-regular graph with diameter \(d \geq 2\). A regular strongly closed subgraph of \(\Gamma\) is said to be a subspace of \(\Gamma\). Define the empty set \(\emptyset\) to be the subspace with diameter \(-1\) in \(\Gamma\). For \(0 \leq i \leq d-1\), let \(\mathcal{L}(\leq i)\) (resp. \(\mathcal{L}(\geq i)\)) denote the set of all subspaces in \(\Gamma\) with diameters \(< i\) (resp. \(\geq i\)) including \(\Gamma\) and \(\emptyset\). If we define the partial order on \(\mathcal{L}(\leq i)\) (resp. \(\mathcal{L}(\geq i)\)) by reverse inclusion (resp. ordinary inclusion), then \(\mathcal{L}(\leq i)\) (resp. \(\mathcal{L}(\geq i)\)) is a poset, denoted by \(\mathcal{L}_R(\leq i)\) (resp. \(\mathcal{L}_o(\geq i)\)). In the present paper, we give the eigenpolynomials of \(\mathcal{L}_R(\leq i)\) and \(\mathcal{L}_o(\geq i)\).

A radio \(k\)-labeling of a connected graph \(G\) is an assignment \(f\) of non-negative integers to the vertices of \(G\) such that

\[|f(x) – f(y)| \geq k + 1 – d(x, y),\]

for any two vertices \(x\) and \(y\), where \(d(x, y)\) is the distance between \(x\) and \(y\) in \(G\). The radio antipodal number is the minimum span of a radio \((diam(G) – 1)\)-labeling of \(G\) and the radio number is the minimum span of a radio \((diam(G))\)-labeling of \(G\).

In this paper, the radio antipodal number and the radio number of the hypercube are determined by using a generalization of binary Gray codes.

In this article, the planes meeting a non-singular quadric of PG\((4,q)\) in a conic are characterized by their intersection properties with points, lines and \(3\)-spaces.

Some Krasnotel’skii-type results previously established for a simply connected orthogonal polygon may be extended to a nonempty compact planar set \(S\) having connected complement. In particular, if every two points of \(S\) are visible via staircase paths from a common point of \(S\), then \(S\) is starshaped via staircase paths. For \(n\) fixed, \(n \geq 1\), if every two points of \(S\) are visible via staircase \(n\)-paths from a common point of \(S\), then \(S\) is starshaped via staircase \((n+1)\)-paths. In each case, the associated staircase kernel is orthogonally convex.