In this paper, we introduce \(h(x)\)-Lucas quaternion polynomials that generalize \(k\)-Lucas quaternion numbers that generalize Lucas quaternion numbers. Also we derive the Binet formula and generating function of \(h(x)\)-Lucas quaternion polynomial sequence.

We determine the crossing numbers (i) of the complete graph \(K_n\) with an edge deleted for \(n \leq 12\) and (ii) of the complete bipartite graph \(K_{m,n}\) with an edge deleted for \(m \in \{3,4\}\) and for all natural numbers \(n$\), and also for the case \(m = n = 5\).

A \(G\)-design is called balanced if the degree of each vertex \(x\) is a constant. A \(G\)-design is called strongly balanced if for every \(i = 1, 2, \ldots, h\), there exists a constant \(C_i\) such that \(d_{A_i}(x) = C_i\) for every vertex \(x\), where \(A_i\) are the orbits of the automorphism group of \(G\) on its vertex-set and \(d_{A_i}(x)\) of a vertex is the number of blocks containing \(x\) as an element of \(A_i\). We say that a \(G\)-design is simply balanced if it is balanced, but not strongly balanced. In this paper, we determine the spectrum for simply balanced and strongly balanced House-systems. Further, we determine the spectrum for House-systems of all admissible indices nesting \(C_4\)-systems.

The Wiener index of a graph is the sum of the distances between all pairs of vertices. In this paper, we determine \(h\)-cacti and \(h\)-cactus chains with the extremal Wiener indices, respectively.

A cyclic coloring is a vertex coloring such that vertices incident with the same face receive different colors. Let \(G\) be a plane graph, and let \(\Delta^*\) be the maximum face degree of \(G\). In 1984, Borodin conjectured that every plane graph admits a cyclic coloring with at most \(\left\lfloor \frac{3\Delta^*}{2} \right\rfloor\) colors. In this note, we improve a result of Borodin et al. [On cyclic colorings and their generalizations, Discrete Mathematics 203 (1999), 23-40] by showing that every plane graph with \(\Delta^* = 6\) can be cyclically colored with 9 colors. This confirms the Cyclic Coloring Conjecture in the case \(\Delta^* = 6\).

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.