In this study, we analyze the number of pairs (i.e., two cards of the same rank) in a set of cards randomly selected from a deck of playing cards. While the standard deck of playing cards comprises 52 cards excluding the joker with 4 suits and 13 ranks, our analysis considers a generalized deck where the numbers of suits and ranks can be set arbitrarily. We derive the exact formulas for the mean and variance of the number of pairs in a given number of cards randomly selected from this generalized deck, expressed using the Gauss hypergeometric function. Moreover, we derive the asymptotic behavior of the mean and variance as the number of ranks tends to infinity.
For a connected graph G, its resistance distance matrix is denoted by R(G). A graph is called resistance regular if all the row (or column) sums of R(G) are equal. We provide a necessary and sufficient condition for a simple connected graph to be resistance regular. Additionally, we establish sharp bounds for the resistance spectral radius and present various bounds for the resistance energy of G. Furthermore, we compute the resistance spectrum and resistance energy of some resistance regular graphs.
In this paper, we introduce a novel graph structure, namely the subdivision-edge-cycle graph, denoted by SE 1 (X), and derive explicit expressions for the first Zagreb index, second Zagreb index and F-index. The chemical relevance of the proposed graph transformation is demonstrated through QSPR modeling of benzenoid hydrocarbons, where it exhibits strong correlations with π-electron energy, molar refractivity, polarizability and exact mass outperforming the corresponding Zagreb indices and F-index of T 1 (X), T 2 (X), and T(X) graphs across linear, quadratic, and cubic regression models. Furthermore, we define a four types of graph operations involving subdivision-edge-cycle graph namely the SE 1 -vertex corona product, SE 1 -edge corona product, SE 1 -vertex neighborhood corona product, and SE 1 -edge neighborhood corona product. For each of these constructions, closed-form expressions for the first Zagreb index, second Zagreb index and F-index are obtained. In addition, bounds for the general sum-connectivity index of these graphs are established.
Tyrosine kinase inhibitors (TKIs) are a type of targeted cancer drugs that work by blocking particular pathways that promote angiogenesis and tumour growth. Although the biochemical mechanisms of TKIs have been well studied, further understanding of their complexity, classification, and possible combinatorial behaviour can be gained by structural graph-theoretic research. In this study, we determine the metric dimension and all possible metric bases of molecular graphs corresponding to selected TKIs such as Sunitinib, Sorafenib, Axitinib, Regorafenib, Cabozantinib, Pazopanib and Lenvatinib. By identifying the metric bases of these molecular structures, we may investigate how structural distinguishability can be used in cheminformatics methods to drug repurposing, molecular categorization, and combination therapy modelling. While the real-world chemical information by considering atoms and bonds as discrete vertices and edges can be simplified by molecular graphs, the analysis of them remains useful in computational drug discovery. This work enables more research at the intersection of discrete mathematics and biomedical science and helps in mathematical characterization of anti-cancer drugs.
The minimum weight k-path vertex cover problem is defined on a vertex-weighted graph G, where the objective is to find a vertex subset S such that every path of order k contains at least one vertex in S, while minimizing the total weight of S. For any integer k ≥ 2, this problem is NP-hard on general graphs. In this study, we focused on the case k = 4. We propose a hybrid framework that integrates a deep Q-network with a local search algorithm. Experimental results on randomly generated instances demonstrate that our method outperforms baseline algorithms and exhibits strong generalization.
We present some properties of Carol numbers. We show the relationship between Carol numbers and Mersenne numbers and also between Carol numbers and other numbers. We also study the relationship between other classes of numbers, sometimes arriving at interesting hypotheses that remain unsolved to this day. We derive the generating function for Carol numbers.
External and internal equitability studied with the properties of either the components or the set itself. In this paper, the equitability is imposed on the individual vertices and their dominating vertices. A subset [Formula: see text] of [Formula: see text] is called a Point wise outward equitable domination if for any [Formula: see text] there exists [Formula: see text] such that [Formula: see text] and [Formula: see text] are adjacent and [Formula: see text] This concept is different from that of degree equitable domination [3,4] studied in [2] where for any vertex [Formula: see text]there exists [Formula: see text] such that [Formula: see text] This is also different from the out degree equitable domination studied in [2] where [Formula: see text] is a dominating set and for any [Formula: see text][Formula: see text] Domination with this condition is the subject matter of study of this paper.
The Zagreb index is a topological index that is defined based on the degree of vertices. In this paper, the concepts of rainbow chromatic Zagreb indices and rainbow chromatic irregularity indices for graphs are discussed. Here, we consider the degree of colors/labels used. The first and second rainbow chromatic Zagreb indices and their irregularity indices of well-known graph classes are calculated. Later, the paper conducts Quantitative structure-property relationship analysis on specific alkanes to demonstrate the practicality of the topological indices outlined.
The concepts of integrity play a crucial role in graph theory. In this study, we introduce a novel parameter called the integrity degree of a graph. The integrity degree of a vertex u is the minimum cardinality of a minimal integrity set that contains u. By using this concept, this study relates several inequalities involving the integrity degree with other established graph parameters. A new class of graphs called dI - k regular graphs are defined. The integrity degree is investigated for various classes of graphs such as complete graphs, paths, cycles, wheel graphs, complete bipartite graphs, book graphs, windmill graphs, star graphs. The computation and illustration of integrity degree of the vertices of Petersen graph is provided. The integrity degree of vertices in binary trees and binomial trees is computed. Also, an application of integrity degree of vertices in communication network is discussed.
In this paper, we study skew polycyclic codes over the ring [Formula: see text] where [Formula: see text] and [Formula: see text] where [Formula: see text] is an odd prime. We investigate the structural properties of skew polycyclic codes over [Formula: see text] through a decomposition theorem. Furthermore, we examine the dual codes of skew polycyclic codes over R, providing necessary and sufficient conditions for a code to be self-dual. We investigate the Gray images of skew polycyclic codes over R, focusing on codes with good parameters. We discuss skew polycyclic codes over the ring [Formula: see text] with even characteristic and provide several illustrative examples to support the theoretical results.
Intrusion Detection Systems (IDSs) are essential for protecting networks from malicious attacks. Though the existing research provided several advancements in the research area, there are various issues, including high false positive rates, delayed detection times, tampering, and inefficiencies in managing dynamic threats that limit their performance. Advanced research works are required because conventional IDS frequently cannot keep up with the growing complexity of infiltration techniques. To overcome these drawbacks and to present a novel model, Platypterus Elite Bubble-net Optimization-enabled Stacked Convolutional Extreme Gradient Boosting, coupled with Parallel Proof of Work (PEBO-CnXGB-PPOW), is presented in the research. PEBO-CnXGB-PPOW makes use of Stacked Convolutional XGBoost to improve intrusion detection speed and accuracy by utilizing strong feature extraction and classification capabilities. To elevate the research, Blockchain-based storage is incorporated as a security concern. By combining the best features of whale and sailfish optimization methods, the PEBO algorithm achieves higher performance by allowing for faster convergence and effective parameter adjustment. For the most part, PEBO-CnXGB-PPOW not only addresses the shortcomings of current IDS but also establishes new standards for security, adaptability, and efficacy in the face of constantly changing cybersecurity threats. The numerical achievements are evaluated with accuracy, precision, and recall achieved 98.36%, 98.91%, and 97.57%, respectively.
A secure set of a graph is a set of vertices that can defend against attacks from the neighborhood of any of its subsets. The security number of a graph is the minimum cardinality of a secure set. In this paper, we obtain a lower bound on the security number of the Cartesian product of two connected graphs. Additionally, we characterize the graphs for which the bound is attained.
An injective [Formula: see text]-edge-coloring of a graph [Formula: see text] is a mapping [Formula: see text]: [Formula: see text], such that [Formula: see text] if edges [Formula: see text] and [Formula: see text] are at distance two, or are in a triangle. The smallest integer [Formula: see text] such that [Formula: see text] has an injective [Formula: see text]-edge-coloring is called the injective chromatic index of [Formula: see text], denoted by [Formula: see text]. In this paper, we prove that [Formula: see text] for every graph [Formula: see text] with [Formula: see text] and mad[Formula: see text], where [Formula: see text] is the maximum degree of [Formula: see text].
In this paper, we construct a family of new pooling designs based on the singular linear spaces and related counting theorems. By analyzing its disjunct properties, it turns out that the new design is superior to the existing designs under the same experimental efficiency. Additionally, we compare our pooling design with Liu et al. with regard to the parameters of construction. At last, the relevant proof of our pooling designs is given.
A graph [Formula: see text] is said to be integral if all its adjacency eigenvalues are integers. For odd [Formula: see text], Cheng, Feng and Liu gave some necessary or sufficient conditions for the integrality of Cayley graph on nonabelian group [Formula: see text] in the paper “T. Cheng et al., Integral Cayley graphs over a certain nonabelian group, Linear Multilinear Algebra 70(21) (2022) 7224–7235”. Meanwhile, it is pointed out that the problem in the aforementioned study may be considered for even [Formula: see text]. Therefore, to provide readers with a complete characterization on the integrality of Cayley graphs over [Formula: see text], we present some necessary or sufficient conditions for the case when [Formula: see text] is even.
Let [Formula: see text] be an integer. The [Formula: see text]-hued chromatic number [Formula: see text] of a graph [Formula: see text] is the minimum [Formula: see text] such that [Formula: see text] admits a proper [Formula: see text]-coloring where each vertex [Formula: see text] has at least [Formula: see text] distinct colors in its neighborhood. In this paper, we prove that if [Formula: see text] is a maximal planar graphs with diameter two, then [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], and [Formula: see text] if [Formula: see text]. These upper bounds are sharp. The result confirms the r-hued coloring conjecture proposed by Song at al. in [Discrete Math. 315 (2014) 47-52] for maximal planar graphs with diameter two.
In this paper, we investigate quadratic residue codes of prime length [Formula: see text] over [Formula: see text] where [Formula: see text] or [Formula: see text] These codes are defined via their generating idempotents, and we examine their extended versions. Furthermore, we demonstrate that the extended quadratic residue codes over [Formula: see text] possess large automorphism groups, a property that can be effectively exploited in decoding algorithms.
To address the complexity of hypergraph-related problems, several authors have proposed alternative structures for representing hypergraphs. Although these new representations may improve the understanding of such problems, their associated computational complexity often remains NP-hard. In addition, some authors have developed algorithms tailored to specific types of hypergraphs. In this study, we introduce a novel representation of simple, undirected hypergraphs using the concept of a reduced graph. This representation preserves the high-level relationships of the original hypergraph while enabling the design of efficient algorithms for specific hypergraph classes. We demonstrate that this structure is particularly useful for computing minimal transversals of minimum cardinality as well as minimal transversals of maximum cardinality. Furthermore, we present a simple algorithm to compute the latter for irredundant hypergraphs.
A cactus is a connected graph in which any two cycles have at most one common vertex, making it a natural extension of trees and unicyclic graphs. In this paper, we investigate the extremal behavior of the Laplacian spectral radius among all cactus graphs with n vertices. It is known that the path graph achieves the minimum Laplacian spectral radius among all n-vertex cacti, we show that certain friendship-like cacti maximize it. We identify the cycle graph and the tadpole graph as attaining the smallest and second smallest values, respectively, among non-tree cacti. This paper extends classical extremal results to the spectral characterization of cactus graphs.
The Slater index of a tournament is the minimum number of arcs that must be reversed in that tournament to make it a total order. Let n be an integer with n >= 5. The minimum value of the Slater index over the indecomposable n-vertex tournaments is [n+1/4] . In a recent paper [H. Belkhechine, C. Ben Salha, R. Romdhane, Indecomposable tournaments with minimum Slater index, Adv. Pure Appl. Math. 17 (2026) 60-77], we characterized the indecomposable n-vertex tournaments with at least five vertices and minimum Slater index, i.e., with Slater index [n+1/4]. In this paper, we asymptotically enumerate these tournaments.
