Let \(s(n, k) = \binom{6k}{3k} \binom{3k}{k} (\binom{3(n-k)}{n-k} / (2n-1) \binom{3n}{n})\). Recently, Guo confirmed a conjecture of \(Z.-W\). Sun by showing that \(s(n, k)\) is an integer for \(k = 0, 1, \ldots, n\). Let \(d = (3n + 2) / \gcd(3n + 2, 2n – 1)\). In this paper, we prove that \(s(n, k)\) is a multiple of the odd part of \(d\) for \(k = 0, 1, \ldots, n\). Furthermore, if \(\gcd(k, n) = 1\), then \(s(n, k)\) is also a multiple of \(n\). We also show that the \(2\)-adic order of \(s(n, k)\) is at least the sum of the digits in the binary expansion of \(3n\).

For any non-trivial abelian group \(A\) under addition, a graph \(G\) is said to be strong \(A\)-magic if there exists a labeling \(f\) of the edges of \(G\) with non-zero elements of \(A\) such that the vertex labeling \(f^+\) defined as \(f^+(v) = \sum f(uv)\) taken over all edges \(uv\) incident at \(v\) is a constant, and the constant is same for all possible values of \(|V(G)|\). A graph is said to be strong \(A\)-magic if it admits strong \(A\)-magic labeling. In this paper, we consider \((\mathbb{Z}_4, +)\) as an abelian group and we prove strong \(\mathbb{Z}_4\)-magic labeling for various graphs and generalize strong \(\mathbb{Z}_{4p}\)-magic labeling for those graphs. The graphs which admit strong \(\mathbb{Z}_{4p}\)-magic labeling are called as strong \(\mathbb{Z}_{4p}\)-magic graphs.

The well-known Mantel’s Theorem states that a graph on \(n\) vertices and \(m\) edges contains a triangle if \(m > \frac{n^2}{4}\). Nosal proved that every graph on \(m\) edges contains a triangle if the spectral radius \(\lambda_1 > \sqrt{m}\), which is a spectral analog of Mantel’s Theorem. Furthermore, by using Motzkin-Straus Inequality, Nikiforov sharpened Nosal’s result and characterized the extremal graphs when the equality holds. Our first contribution in this note is to give two new proofs of the spectral concise Mantel’s Theorem due to Nikiforov (without help of Motzkin-Straus Inequality). Nikiforov also obtained some results concerning the existence of consecutive cycles and spectral radius. Second, we prove a theorem concerning the existence of consecutive even cycles and spectral radius, which slightly improves a result of Nikiforov. At last, we focus on spectral radius inequalities. Hong proved his famous bound for spectral radius. Later, Hong, Shu, and Fang generalized Hong’s bound to connected graphs with given minimum degree. By using quite different techniques, Nikiforov proved Hong et al.’s bound for general graphs independently. In this note, we prove a new spectral inequality by applying the technique of Nikiforov. Our result extends Stanley’s spectral inequality.

The alliance polynomial of a graph with order \(n\) and maximum degree \(\Delta\) is the polynomial \(A(\Gamma; x) = \sum_{k=-\delta_1}^{\delta_1}A_k(\Gamma) x^{n+k}\), where \(A_k(G)\) is the number of exact defensive \(k\)-alliances in \(G\). We provide an algorithm for computing the alliance polynomial. Furthermore, we obtain some properties of \(A(\Gamma; x)\) and its coefficients. In particular, we prove that the path, cycle, complete, and star graphs are characterized by their alliance polynomials. We also show that the alliance polynomial characterizes many graphs that are not distinguished by other usual polynomials of graphs.

Let \(a\), \(b\), and \(k\) be three nonnegative integers with \(a \geq 2\) and \(b \geq a(k+1)+2\). A graph \(G\) is called a \(k\)-Hamiltonian graph if \(G – U\) contains a Hamiltonian cycle for every subset \(U \subseteq V(G)\) with \(|U| = k\). An \([a, b]\)-factor \(F\) of \(G\) is called a Hamiltonian \([a, b]\)-factor if \(F\) contains a Hamiltonian cycle. If \(G – U\) has a Hamiltonian \([a, b]\)-factor for every subset \(U \subseteq V(G)\) with \(|U| = k\), then we say that \(G\) admits a \(k\)-Hamiltonian \([a, b]\)-factor. Suppose that \(G\) is a \(k\)-Hamiltonian graph of order \(n\) with \(n \geq a+k+2\). In this paper, it is proved that \(G\) includes a \(k\)-Hamiltonian \([a, b]\)-factor if \(\delta(G) \geq a+k\) and \(t(G) \leq a-1+\frac{(a-1)(k+1)}{b-2}\).

Graph embedding has been known as a powerful tool for implementation of parallel algorithms or simulation of different interconnection networks. An embedding \(f\) of a guest graph \(G\) into a host graph \(H\) is a bijection on the vertices such that each edge of \(G\) is mapped into a path of \(H\). In this paper, we introduce a graph called the generalized book and the main results obtained are: (1) For \(r \geq 3\), the minimum wirelength of embedding \(r\)-dimensional hypercube \(Q_r\) into the generalized book \(\mathrm{GB}[2^{r_1}, 2^{r_2}, 2^{r_3}]\), where \(r_1 + r_2 + r_3 = r\). (2) A linear time algorithm to compute the exact wirelength of embedding hypercube into generalized book. (3) An algorithm for embedding hypercube into generalized book with dilation 3, proving that the lower bound obtained by Manuel et al. [28] is sharp.

In this paper, we present a new approach to the convolved Fibonacci numbers arising from the generating function of them and give some new and explicit identities for the convolved Fibonacci numbers.

The generalized Fibonacci cube \(Q_d(f)\) is the graph obtained from the hypercube \(Q_d\) by removing all vertices that contain a given binary word \(f\). A binary word \(f\) is called good if \(Q_d(f)\) is an isometric subgraph of \(Q_d\) for all \(d \geq 1\), and bad otherwise. A non-extendable sequence of contiguous equal digits in a word \(f\) is called a block of \(f\). The question to determine the good (bad) words consisting of at most three blocks was solved by Ilié, Klavžar, and Rho. This question is further studied in the present paper. All the good (bad) words consisting of four blocks are determined completely, and all bad \(2\)-isometric words among consisting of at most four blocks words are found to be \(1100\) and \(0011\).

In this paper, we provide a construction of \(\mathrm{PG}(2,4)\) by a collage of \(\mathrm{AG}(2,3)\) and its dual \(\mathrm{DAG}(2,3)\). Moreover, we prove that the construction is unique.

In this paper, we first present a combinatorial proof of the recurrence relation about the number of the inverse-conjugate compositions of \(2n+1\), \(n > 1\). And then we get some counting results about the inverse-conjugate compositions for special compositions. In particular, we show that the number of the inverse-conjugate compositions of \(4k+1\), \(k > 0\) with odd parts is \(2^k\), and provide an elegant combinatorial proof. Lastly, we give a relation between the number of the inverse-conjugate odd compositions of \(4k+1\) and the number of the self-inverse odd compositions of \(4k+1\).