In this paper, we give some new identities of symmetry for \(q\)-Bernoulli polynomials under the symmetric group of degree \(n\) arising from \(p\)-adic \(q\)-integrals on \(\mathbb{Z}_p\).

The covering and packing of a unit square (resp. cube) with squares (resp. cubes) are considered. In \(d\)-dimensional Euclidean space \(\mathbb{E}^d\), the size of a \(d\)-hypercube is given by its side length and the size of a covering is the total size of the \(d\)-hypercubes used to cover the unit hypercube. Denote by \(g_d(n)\) the smallest size of a minimal covering (consisting of \(n\) hypercubes) of a \(d\)-dimensional unit hypercube. In this paper, we consider the problem of covering a unit hypercube with hypercubes in \(\mathbb{E}^d\) for \(d \geq 4\) and determine the tight upper bound and lower bound for \(g_d(n)\).

Given the binomial transforms \(\{b_n\}\) and \(\{c_n\}\) of the sequences \(\{a_n\}\) and \(\{d_n\}\) correspondingly, we compute the binomial transform of the sequence \(\{a_nc_n\}\) in terms of \(\{b_n\}\) and \(\{d_n\}\). In particular, we compute the binomial transform of the sequences \(\{n{n-1}\ldots (n-1-m)a_n\}\) and \(\{a_k x^k\}\) in terms of \(\{b_n\}\). Further applications include new binomial identities with the binomial transforms of the products \(H_n B_n\), \(H_n F_n\), \(H_n L_n(X)\), and \(B_n F_n\), where \(H_n\), \(B_n\), \(F_n\), and \(L_n(X)\) are correspondingly the harmonic numbers, the Bernoulli numbers, the Fibonacci numbers, and the Laguerre polynomials.

A kite graph is a graph obtained from a \(3\)-cycle (or triple) by adding a pendent edge to a vertex of the \(3\)-cycle. A kite system of order \(v\) is a pair \((X, \mathcal{B})\), where \(\mathcal{B}\) is an edge-disjoint collection of kite graphs which partitions the edge set of \(K_v\). A kite system of order \(v\) is cyclic if it admits an automorphism of order \(v\), and 1-rotational if it admits an automorphism containing one fixed point and a cycle of length \(v – 1\). In this paper, we show that there exists a cyclic kite system of order \(v\) if and only if \(v \equiv 1 \pmod{8}\), and there exists a \(1\)-rotational kite system of order \(v\) if and only if \(v \equiv 0 \pmod{8}\).

A \(2\)-rainbow dominating function (2RDF) on a graph \(G = (V, E)\) is a function \(f\) from the vertex set \(V\) to the set of all subsets of the set \(\{1,2\}\) such that for any vertex \(v \in V\) with \(f(v) = \emptyset\) the condition \(\cup_{u \in N(v)} f(u) = \{1, 2\}\) is fulfilled. The weight of a 2RDF \(f\) is the value \(w(f) = \sum_{v \in V(G)} |f(v)|\). The \(2\)-rainbow domination number, denoted by \(\gamma_{r2}(G)\), is the minimum weight of a 2RDF on \(G\). The rainbow bondage number \(b_{r2}(G)\) of a graph \(G\) with maximum degree at least two, is the minimum cardinality of all sets \(E’ \subseteq E(G)\) for which \(\gamma_{r2}(G – E’) > \gamma_{r2}(G)\). Dehgardi, Sheikholeslami, and Volkmann [Discrete Appl. Math. \(174 (2014), 133-139]\) proved that the rainbow bondage number of a planar graph does not exceed 15. In this paper, we improve this result.

Let \(id(v)\) denote the implicit degree of a vertex \(v\) in a graph \(G\). We define \(G\) to be implicit claw-heavy if every induced claw of \(G\) has a pair of nonadjacent vertices such that their implicit degree sum is more than or equal to \(|V(G)|\). In this paper, we show that an implicit claw-heavy graph \(G\) is hamiltonian if we impose certain additional conditions on \(G\) involving numbers of common neighbors of some specific pair of nonadjacent vertices, or forbidden induced subgraphs. Our results extend two previous theorems of Chen et al. [B. Chen, 8. Zhang and S. Qiao, Hamilton cycles in claw-heavy graphs, Discrete Math., \(309 (2009) 2015-2019.]\) on the existence of Hamilton cycles in claw-heavy graphs.

A connected graph \(G\) is called a quasi-tree graph, if there exists \(v_0 \in V(G)\) such that \(G – v_0\) is a tree. In this paper, among all triangle-free quasi-tree graphs of order \(n\) with \(G – v_0\) being a tree and \(d(v_0) = d(v_0)\), we determine the maximal and the second maximal signless Laplacian spectral radii together with the corresponding extremal graphs. By an analogous manner, we obtain similar results on the spectral radius of triangle-free quasi-tree graphs.

The notation \(tK_3\) represents a graph with \(t\) copies of the complete graph \(K_3\). In this note, we discuss the goodness of path \(P_n\) or cycle \(C_n\) with respect to \(tK_3\). Furthermore, this result provides the computation of Ramsey number \(R(G, tK_3)\) when \(G\) is a set of disjoint paths or cycles.

A \(k\)-king in a digraph \(D\) is a vertex which can reach every other vertex by a directed path of length at most \(k\). Every tournament with no vertex of in-degree zero has at least three \(2\)-kings. In this paper, we present the structure of tournaments which have exactly three \(2\)-kings and prove that every strong tournament, containing at least \(k+2\) vertices with \(k \geq 3\), has at least \(k+1\) \(k\)-kings.