In this paper, we derive some identities involving Genocchi polynomials and numbers. These identities follow by evaluating a certain integral in various ways. Also, we express the product of two Genocchi polynomials as a linear combination of Bernoulli polynomials.

Fuzzy graph theory is finding an increasing number of applications in modeling real-time systems where the level of information inherent in the system varies with different levels of precision. Fuzzy models are becoming useful because of their aim in reducing the differences between the traditional numerical models used in engineering and sciences, and the symbolic models used in expert systems. A bipolar fuzzy model is a generalized soft computing model of a fuzzy model that gives more precision, flexibility, and compatibility to a system when compared with systems designed using fuzzy models. In this research article, we introduce certain types of bipolar fuzzy competition graphs, including bipolar fuzzy \(k\)-competition, bipolar fuzzy \(p\)-competition, and bipolar fuzzy \(m\)-competition. We investigate some properties of these new concepts.

The \(\alpha\)-incidence energy of a graph is defined as the sum of \(a\)th powers of the signless Laplacian eigenvalues of the graph, where \(a\) is a real number such that \(\alpha \neq 0\) and \(\alpha \neq 1\). The \(\alpha\)-distance energy of a graph is defined as the sum of \(a\)th powers of the absolute values of the eigenvalues of the distance matrix of the graph, where \(\alpha\) is a real number such that \(\alpha \neq 0\). In this note, we present some bounds for the \(\alpha\)-incidence energy of a graph. We also present some bounds for the \(\alpha\)-distance energy of a tree.

Multi-sender authentication codes allow a group of senders to construct an authenticated message for a receiver such that the receiver can verify authenticity of the received message. In this paper, we construct one multi-sender authentication codes from
polynomials over finite fields. Some parameters and the probabilities of deceptions of this codes are also computed.

A graph \(G\) is called \((k, d)^*\)-choosable if for every list assignment \(L\) satisfying \(|L(v)| \geq k\) for all \(v \in V(G)\), there is an \(L\)-coloring of \(G\) such that each vertex of \(G\) has at most \(d\) neighbors colored with the same color as itself. In this paper, it is proved that every graph of nonnegative characteristic without \(4\)-cycles and intersecting triangles is \((3, 1)^*\)-choosable.

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.