In this paper, we study \((2-d)\)-kernels in graphs. We shall show that the problem of the existence of \((2-d)\)-kernels is \(\mathcal{N}P\)-complete for a general graph. We also give some results related to the problem of counting \((2-d)\)-kernels in graphs. For special graphs, we show that the number of \((2-d)\)-kernels is equal to the Fibonacci numbers.

In 1989, Frankl and Füredi [1] conjectured that the \(r\)-uniform hypergraph with \(m\) edges formed by taking the first \(m\) sets in the colex ordering of \(\mathbb{N}^{(r)}\) has the largest Lagrangian of all \(r\)-uniform hypergraphs of size \(m\). For \(2\)-graphs, the Motzkin-Straus theorem implies this conjecture is true. For \(3\)-uniform hypergraphs, it was proved by Talbot in 2002 that the conjecture is true while \(m\) is in a certain range. In this paper, we prove that the \(4\)-uniform hypergraphs with \(m\) edges formed by taking the first \(m\) sets in the colex ordering of \(\mathbb{N}^{(r)}\) has the largest Lagrangian of all \(4\)-uniform hypergraphs with \(t\) vertices and \(m\) edges satisfying \(\binom{t-1}{4} \leq m \leq \binom{t-1}{4} + \binom{t-2}{3} – 17\binom{t-2}{2} + 1\).

A graph \(G\) on \(n \geq 3\) vertices is called claw-heavy if every induced claw of \(G\) has a pair of nonadjacent vertices such that their degree sum is at least \(n\). We say that a subgraph \(H\) of \(G\) is \(f\)-heavy if \(\max\{d(x), d(y)\} \geq \frac{n}{2}\) for every pair of vertices \(x, y \in V(H)\) at distance \(2\) in \(H\). For a given graph \(R\), \(G\) is called \(R\)-\(f\)-heavy if every induced subgraph of \(G\) isomorphic to \(R\) is \(f\)-heavy. For a family \(\mathcal{R}\) of graphs, \(G\) is called \(\mathcal{R}\)-\(f\)-heavy if \(G\) is \(R\)-\(f\)-heavy for every \(R \in \mathcal{R}\). In this paper, we show that every \(2\)-connected claw-heavy graph is hamiltonian if \(G\) is \(\{P_7, D\}\)-\(f\)-heavy, or \(\{P_7, H\}\)-\(f\)-heavy, where \(D\) is a deer and \(H\) is a hourglass. Our result is a common generalization of previous theorems of Broersma et al. and Fan on hamiltonicity of \(2\)-connected graphs.

An \(H_3\) graph is a multigraph on three vertices with double edges between two pairs of distinct vertices and a single edge between the third pair. In this paper, we decompose a complete multigraph \(2K_{10t}\) into \(H_3\) graphs.

In 1989, Zhu, Li, and Deng introduced the definition of implicit degree, denoted by \(\text{id}(v)\), of a vertex \(v\) in a graph \(G\). In this paper, we give a simple method to prove that: if \(G\) is a \(k\)-connected graph of order \(n\) such that the implicit degree sum of any \(k+1\) independent vertices is more than \((k+1)(n-1)/2\), then \(G\) is hamiltonian. Moreover, we provide an algorithm according to the proof.

Let \(D\) be a finite and simple digraph with vertex set \(V(D)\), and let \(f: V(D) \to \{-1, 1\}\) be a two-valued function. If \(\sum_{x \in N_D^-[v]} f(x) \geq 1\) for each \(v \in V(D)\), where \(N_D^-[v]\) consists of \(v\) and all vertices of \(D\) from which arcs go into \(v\), then \(f\) is a signed dominating function on \(D\). The sum \(\sum_{v \in V(D)} f(v)\) is called the weight of \(f\). The signed domination number, denoted by \(\gamma_S(D)\), of \(D\) is the minimum weight of a signed dominating function on \(D\). In this work, we present different lower bounds on \(\gamma_S(D)\) for general digraphs, show that these bounds are sharp, and give an improvement of a known lower bound obtained by Karami in 2009 [H. Karami, S.M. Sheikholeslami, A. Khodkar, Lower bounds on the signed domination numbers of directed graphs, Discrete Math. 309 (2009), 2567-2570]. Some of our results are extensions of well-known properties of the signed domination number of graphs.

Let \(G\) be a graph of order at least \(2k\) and \(s_1, s_2, \ldots, s_k, t_1, t_2, \ldots, t_k\) be any \(2k\) distinct vertices of \(G\). If there exist \(k\) disjoint paths \(P_1, P_2, \ldots, P_k\) such that \(P_i\) is an \(s_i – t_i\) path for \(1 \leq i \leq k\), we call \(G\) \(k\)-linked. K. Kawarabayashi et al. showed that if \(n \geq 4k – 1\) (\(k \geq 2\)) with \(\sigma_2(G) \geq n + 2k – 3\), then \(G\) is \(k\)-linked. Li et al. showed that if \(G\) is a graph of order \(n \geq 232k\) with \(\sigma_2^*(G) \geq n + 2k – 3\), then \(G\) is \(k\)-linked. For sufficiently large \(n\), it implied the result of K. Kawarabayashi et al. The main purpose of this paper is to lower the bound of \(n\) in the result of Li et al. We show that if \(G\) is a graph of order \(n \geq 111k + 9\) with \(\sigma_2^*(G) \geq n + 2k – 3\), then \(G\) is \(k\)-linked. Thus, we improve the order bound to \(111k + 9\), and when \(n \geq 111k + 9\), it implies the result of \(K\). Kawarabayashi \(et al\).

The classification of all dihedral f-tilings of the Riemannian sphere \(S^2\) ,whose prototiles are two right triangles with at least one isosceles, is given.The combinatorial structure and the symmetry group of each tiling is also achieved.

In [4], the author introduced a new metric on the space \(\text{Mat}_{m \times s}(\mathbb{Z}_q)\), which is the module space of all \(m \times s\) matrices with entries from the finite ring \(\mathbb{Z}_q\) (\(q \geq 2\)), generalizing the classical Lee metric [5] and the array RT-metric [8], and named this metric as GLRTP-metric, which is further renamed as LRTJ-metric (Lee-Rosenbloom-Tsfasman-Jain Metric) in [1]. In this paper, we introduce a complete weight enumerator for codes over \(\text{Mat}_{m \times s}(\mathbb{Z}_q)\) endowed with the LRTJ-metric and obtain a MacWilliams-type identity with respect to this new metric for the complete weight enumerator.

The Zagreb indices and the modified Zagreb indices are important topological indices in mathematical chemistry. In this paper we study the relationship between the modified Zagreb indices and the reformulated modified Zagreb indices with respect to trees.