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.

Incorporating the concept of the scattering number and the idea of the vertex-neighbor-connectivity, we introduce a new graph parameter called the vertex-neighbor-scattering number, which measures how easily a graph can be broken into many components with the removal of the neighborhoods of few vertices, and discuss some properties of this parameter. Some tight upper and lower bounds for
this parameter are also given.

This paper is an extension of the work [On the norms of circulant matrices with the Fibonacci and Lucas numbers, Appl. Math.
and Comp., \(160 (2005), 125-132.]\), in which for some norms of the circulant matrices with classical Fibonacci and Lucas numbers it is
obtained the lower and upper bounds. In this new paper, we generalize the results of that work.

Let \(a_0, a_1, \ldots, a_{r-1}\) be positive integers and define a conditional sequence \(\{q_n\}\), with initial conditions \(q_0 = 0\) and \(q_1 = 1\), and for all \(n \geq 2\), \(q_n = a_1q_{n-1} + q_{n-2}\) where \(n \equiv t \pmod{r}\). For \(r = 2\), the author studied it in \([1]\). For general \(\{q_n\}\), we found a closed form of the generating function for \(\{q_n\}\) in terms of the continuant in \([2]\). In this paper, we give the matrix representation and a Binet-like formula for the conditional sequence \(\{q_n\}\) by using the matrix methods.

A locally \(nK_2\) graph \(G\) is a graph such that the set of neighbors of any vertex of \(G\) induces a subgraph isomorphic to \(nK_2\). We show that a locally \(nK_2\) graph \(G\) must have at least \(6n – 3\) vertices, and that a locally \(nK_2\) graph with \(6n – 3\) vertices exists if and only if \(n \in \{1, 2, 3, 5\}\), and in these cases the graph is unique up to isomorphism. The case \(n = 5\) is surprisingly connected to a classic theorem of algebraic geometry: The only locally \(5K_2\) graph on \(6 \times 5 – 3 = 27\) vertices is the incidence graph of the 27 straight lines on any nonsingular complex projective cubic surface.