Note on the spectra of signed complete bipartite graphs with regular negative edges

Degree-based graph entropies quantify structural heterogeneity by transforming vertex-degree information into a probability distribution and applying Shannon entropy. We develop a unified framework for three Prodeg-type degree-power invariants, namely the Inverse Prodeg index [Formula: see text], the Misbalance Prodeg index [Formula: see text], and the Yemen Prodeg index [Formula: see text], together with their associated entropies [Formula: see text], [Formula: see text], and [Formula: see text]. More generally, for [Formula: see text] we consider the degree-weighted distribution [Formula: see text] and the Shannon entropy [Formula: see text], recovering the Prodeg cases at [Formula: see text]. We derive closed-form expressions for representative graph families (complete graphs, cycles, paths, stars, and complete bipartite graphs) and establish sharp extremal behavior: for connected graphs on [Formula: see text] vertices, [Formula: see text], with equality if and only if G is regular, while highly imbalanced families (e.g., stars) exhibit strong concentration and vanishing entropy for [Formula: see text] as [Formula: see text]. We further provide explicit two-sided bounds in terms of degree extremes and concentration control via the heaviest weight. A main structural result is a tensor-product principle: [Formula: see text] is multiplicative under the tensor (Kronecker) product, implying additivity of [Formula: see text] and the Nordhaus-Gaddum-type bound [Formula: see text] whenever both entropies are defined. Using majorization, we also prove a monotone exponent hierarchy [Formula: see text], with equality throughout precisely for regular graphs. To demonstrate chemical relevance, we analyze [Formula: see text] antibacterial compounds curated from the ChEMBL database and show that both classical degree-entropies and Prodeg entropies strongly track established molecular information/complexity measures (BertzCT and [Formula: see text]), while AvgIpc exhibits weaker and nonlinear associations. Finally, we benchmark entropy-only QSPR models for nine physicochemical endpoints using 5-fold cross-validation. Tree ensembles deliver the strongest performance, with particularly high accuracy for size-related properties such as MolMR ([Formula: see text]) and Molecular Weight ([Formula: see text]), whereas MolLogP remains challenging ([Formula: see text]). Across endpoints, the Prodeg block is competitive despite using fewer descriptors, and combining classical and Prodeg entropies yields consistent (typically modest) RMSE gains, supporting Prodeg entropies as compact and interpretable descriptors for chemical graph analysis and QSPR modeling.

A topological index is a numerical invariant of a graph, typically defined in terms of the degrees or distances of its vertices, and is widely used in QSAR and QSPR studies. In this paper, we introduce the diminished downhill Sombor index together with its exponential variant. Exact values of the proposed index are determined for regular graphs and several standard graph classes, including cycle graphs, complete graphs, path graphs, complete bipartite graphs, and star graphs. Corresponding closed-form expressions for the exponential are also derived. Furthermore, sharp upper and lower bounds for the diminished downhill Sombor index are established.

A spectral stability result regarding the complete bipartite graph K2,

Graph is a discrete mathematical representation widely used to model connectivity in various fields. This study discusses the application of Hamiltonian concepts on the complete bipartite graph K_8^2 as a model for vaccine service route planning at the local level (district). The graph K_8^2 represents eight health facilities in a region, where each facility consists of seven service locations and one other supporting facility. The research method includes constructing a complete bipartite graph, identifying vertices and circuits, and tracing Hamiltonian paths to visualize routes that connect each vertex once in a closed journey. The results show that the K_8^2 graph is effective for designing vaccine distribution routes that systematically and efficiently connect all facilities across eight districts. The complete bipartite graph with Hamiltonian properties proves applicable as a model for public service route planning, with potential implementation in optimizing health logistics distribution, facility inspection, and emergency response for vaccination services at the local level.

This article develops a unified framework for the redefined Zagreb descriptors that combines graph theory, spectral analysis, and molecular structure-property applications. We study the basic redefined Zagreb descriptors together with their higher-order variants, introduce the associated weighted graph matrices, and investigate their spectral radii, energies, and related structural interpretations. For several standard graph families, including paths, cycles, complete graphs, stars, complete bipartite graphs, wheels, and friendship graphs, we derive explicit formulas and show how these descriptors reflect different degree patterns and connectivity structures. We also establish general bounds for the descriptors and their weighted spectral quantities, thereby clarifying their connections with classical degree-based indices and adjacency energy. To examine chemical relevance, we first consider a small set of anticancer drug-like molecules and use it as an analytical descriptor study. In this setting, the redefined Zagreb descriptors and their energy-based analogues are strongly associated with size-related physicochemical quantities, while the mixed higher-order descriptor [Formula: see text] shows the strongest relationship with the minimum universal force-field energy. This part of the study is intended to identify informative descriptor trends rather than to establish a fully validated predictive model. We then carry out a broader QSPR study on 100 alcohol compounds with 17 physicochemical endpoints under repeated leakage-safe grouped external validation. The results show that the redefined Zagreb descriptor family and its derivative forms provide strong predictive performance for many targets, especially those related to molecular size, volume, and critical-property behavior. The derivative descriptors are therefore chemically meaningful and useful, while the combined representation shows where complementary information can be gained. Overall, the redefined Zagreb framework emerges as a mathematically rich and chemically useful family of descriptors whose combinatorial, spectral, and predictive roles can be studied in a unified way.

ABSTRACT The connectivity of a network is directly related to its fault tolerance. The H -structure connectivity is a generalization of the classical connectivity of a graph. In this paper, we give a tight upper bound on the H -structure connectivity of a graph and obtain the precise values of the path-structure and star-structure connectivity of complete bipartite graphs. For Cartesian product graphs, lexicographic product graphs and complete product graphs, the relationship between the structure connectivity of the product graphs and the structure connectivity of the factor graphs is explored. These results are more universal and make the contents of the structure connectivity richer in terms of product graphs.

An edge-based assortativity coefficient is introduced to quantify the strength of degree interaction across edges in a finite simple graph. Unlike correlation-based assortativity measures, the proposed coefficient is defined in terms of the mean of vertex degree sums over edges and is shown to be strictly positive for every nontrivial graph. Fundamental properties are established, including the sharp bounds $0<r(G) \le 1$ and the characterization of regular graphs as those attaining the upper bound. Explicit expressions are derived for several classical families of graphs, including paths, cycles, stars, wheels, friendship graphs, complete graphs, and complete bipartite graphs. These results provide a theoretical framework for analyzing degree interaction strength in graphs and highlight structural features captured by the proposed assortativity measure.

Graphs are simple structures that are easy to understand and fathom. Graphs can be classified into different types based on their structure one such type is the bipartite graph. A nigh complete bipartite graph or a nigh graph is an adaptation of a bipartite graph that is defined as Km, n + e1 + e2 such that the graph becomes a complete bipartite graph when the two edges e1 and e2 are removed. In this paper, the approval of vertex antimagic edge slither labeling to nigh complete bipartite graph is established.

Graphs containing self-loops provide a versatile framework for modeling heteroatomic molecules, with each self-loop representing a hetero-atom. In this study, we investigate the predictive capability of the first Zagreb energy in relation to the physicochemical properties of kidney infection drugs, using their corresponding molecular graphs with self-loops. The analysis using linear, quadratic, cubic, and logarithmic regression models reveals a strong correlation between the first Zagreb energy and key physicochemical properties such as polarizability, molar refractivity, and molar volume. Statistical metrics such as standard error (SE), F-test value, standard error of fit (SF), and root mean squared error (RMSE) validate the stability and reliability of the proposed models. Furthermore, we compute the first Zagreb energy of the complete graph [Formula: see text], as well as the complete bipartite graph [Formula: see text], with partite sets [Formula: see text], and N. In addition, we derive both lower and upper bounds for the first Zagreb energy of graphs with self-loops.

Fix m≥3. The choice number ch(Km,n) of the complete bipartite graph Km,n has two sharp thresholds as n grows. We give complete proofs of the Hoffman–Johnson values at levels m+1 and m, and we pin down the extremal list assignments at the lower threshold n0=(m−1)m−1−(m−2)m−1. Specifically, ch(Km,n)=m+1,n≥mm,m,n0≤n<mm,≤m−1,n<n0. Our method centers on a transversal obstruction principle and a dichotomy for how the M-side lists can intersect when all lists have size m−1: Case I, in which some m−1 of the M-lists are pairwise disjoint, and Case II, in which three M-lists pairwise intersect with all remaining lists mutually disjoint. For m≥5 we show that the three-way intersection pattern (three pairwise intersecting M-lists) is strictly non-extremal, and we prove the uniqueness of extremal configurations: we classify all uniformly critical assignments at n0 and show that, up to relabeling, there are exactly two extremal types for m≥5, while a third type appears for m∈{3,4}. Finally, we propose a fixed-k block model for deeper levels ch(Km,n)=m−k+1 and contrast this unbalanced setting with the balanced case m=n, where ch(Km,m)∼log2m, highlighting the shift from polynomial to logarithmic threshold growth.

Abstract In this paper, we present an efficient stable set polynomial of the complete bipartite graph algorithm (SPBA) to address the time-dependent free-surface behavior resulting from unsteady ship movements. Analytical expressions for the horizontal and vertical derivatives are derived using the transient free-surface Green's function (TFSGF). The fundamental concept of the suggested approach is to convert fourth-order nonlinear differential equations into a system of algebraic equations through an operational matrix of derivatives, employing an optimized choice of collocation points in order to achieve precise analytical solutions. Results are compared against the shifted Legendre wavelet method (SLWM) and analytical solutions, illustrating that SPBA produces similar accuracy to SLWM with smoother error growth, reduced computational cost, and slightly improved long-term stability, making it more suitable for large-scale simulations. Eigenvalue spectrum analysis further confirms SPBA's superior long-term accuracy and enhanced stability over SLWM. Also, SPBA has been used to calculate nondimensional added mass and damping coefficients for the zero-velocity radiation problem of a floating hemisphere. Overall, SPBA proves to be a reliable, flexible, and computationally attractive approach.

Chemical graph theory is an interdisciplinary field that integrates fuzzy graph theory and computational methods to model molecular structures as fuzzy graphs and to address associated mathematical problems. Topological indices assign numerical invariants to network structures; among them, the Sombor index, originally introduced in chemical graph theory, provides a useful measure for quantifying the structure of molecular graphs within a fuzzy graph framework. In this paper, we investigate bounds for the Sombor index across several graph families and operations, including edge addition and deletion, broom graphs, fuzzy star graphs on [Formula: see text] vertices, complete bipartite fuzzy graphs, and the star graph ([Formula: see text]). We also examine the relationship between the Sombor index and various properties of alkanes and octane isomers. Furthermore, we demonstrate a significant correlation between this index and multiple thermodynamic parameters, such as heat of vaporization, entropy, acentric factor, and enthalpy of vaporization, while observing a relatively weak correlation with the heat capacity of octane isomers. Finally, we apply the Sombor index in fuzzy graphs ([Formula: see text]) to identify and rank Indian states according to their crime rates. These results highlight the broad applicability of the Sombor index in both chemical graph theory and real-world network analysis.

A Hadamard–Hitchcock decomposition of a multidimensional array is a decomposition that expresses the latter as a Hadamard product of several tensor rank decompositions. Such decompositions can encode probability distributions that arise from statistical graphical models associated to complete bipartite graphs with one layer of observed random variables and one layer of hidden ones, usually called restricted Boltzmann machines. We establish generic identifiability of Hadamard–Hitchcock decompositions by exploiting the reshaped Kruskal criterion for tensor rank decompositions. An algorithm leveraging existing decomposition algorithms for tensor rank decomposition is introduced for computing a Hadamard–Hitchcock decomposition. Numerical experiments illustrate its computational performance and numerical accuracy in a noiseless setting.

In the k-diameter component connectivity model a network is consider operational if there is a component with diameter at least k. Therefore, a network is in a failure state if every component has diameter less than k. In this paper we find the vertex variant of the k-diameter component connectivity parameter, which is the minimum number of vertex deletions in order to put a network into a failure state, for particular classes of graphs. We also show the mixed variant by allowing vertex and edge failures within the network. We show results for paths, cycles, complete, and complete bipartite graphs for both variants as well as perfect r-ary trees for the vertex variant.

An innovative graph polynomial numerical strategy for the time-fractional Kuramoto-Sivashinsky model based on the complete bipartite graph's independence polynomials

Abstract Biclustering, also known as co-clustering or two-way clustering, simultaneously partitions the rows and columns of a data matrix to reveal submatrices with coherent patterns. Incorporating background knowledge into clustering to enhance solution quality and interpretability has attracted growing interest in mathematical optimization and machine learning research. Extending this paradigm to biclustering enables prior information to guide the joint grouping of rows and columns. We study constrained biclustering with pairwise constraints, namely must-link and cannot-link constraints, which specify whether objects should belong to the same or different biclusters. As a model problem, we address the constrained version of the k-densest disjoint biclique problem, which aims to identify k disjoint complete bipartite subgraphs (called bicliques) in a weighted complete bipartite graph, maximizing the total density while satisfying pairwise constraints. We propose both exact and heuristic algorithms. The exact approach is a tailored branch-and-cut algorithm based on a low-dimensional semidefinite programming (SDP) relaxation, strengthened with valid inequalities and solved in a cutting-plane fashion. Exploiting integer programming tools, a rounding scheme converts SDP solutions into feasible biclusterings at each node. For large-scale instances, we introduce an efficient heuristic based on the low-rank factorization of the SDP. The resulting nonlinear optimization problem is tackled with an augmented Lagrangian method, where the subproblem is solved by decomposition through a block-coordinate projected gradient algorithm. Extensive experiments on synthetic and real-world datasets show that the exact method significantly outperforms general-purpose solvers, while the heuristic achieves high-quality solutions efficiently on large instances.

Graph-theoretic degree-based descriptors play a central role in chemoinformatics and QSPR/QSAR modelling, yet most classical indices either focus purely on vertex degrees or treat bond contributions in a purely multiplicative way. In this work we introduce and systematically study a new family of modified bond-based indices in which each edge [Formula: see text] is weighted by a local bond factor [Formula: see text] in the denominator, coupled with a vertex kernel in the numerator. This construction yields modified versions of the first and second Zagreb indices, the Forgotten and Yemen indices, several connectivity-type descriptors (product, sum, Nirmala, ABC, CAB, GA, harmonic, and misbalance prodeg), as well as Sombor- and Dharwad-type bond indices. We first present a unified edge-partition representation for any symmetric kernel, expressing each modified index as a finite sum over degree classes [Formula: see text]. This framework allows us to derive closed-form expressions for all sixteen modified bond-based indices on a broad collection of benchmark families: paths [Formula: see text], cycles [Formula: see text], complete graphs [Formula: see text], complete bipartite graphs [Formula: see text], stars [Formula: see text], friendship graphs [Formula: see text], wheels [Formula: see text], book graphs [Formula: see text], Dutch windmill graphs [Formula: see text], and hypercubes [Formula: see text]. The resulting tables reveal clear asymptotic growth patterns and highlight which structures are extremal for the modified descriptors. Moreover, we obtain sharp degree-extreme bounds for a representative subset of the indices in terms of the order [Formula: see text], size m, and the minimum and maximum degrees δ and Δ, with equality characterizing regular graphs. The proposed modified bond-based indices thus provide a flexible and analytically tractable family of descriptors that couple vertex and bond information in a novel way, and are well suited as structured features for modern chemoinformatics and graph-based machine-learning models on molecular graphs. Finally, to demonstrate predictive utility in a hypothesis-driven setting, we further benchmark these [Formula: see text] descriptors within a large multi-task QSAR/QSPR pipeline on 3,219 ChEMBL antibacterial molecules across ten continuous properties using a heterogeneous model zoo under three descriptor scenarios, where the combined descriptors scenario achieves the best overall generalisation (Macro Test [Formula: see text]; Global zRMSE [Formula: see text]), improving upon the Physicochemical descriptors scenario (Macro Test [Formula: see text]; Global zRMSE [Formula: see text]).

Let L be a finite lattice with two atoms. In this paper we have shown existence of sublattice N such that N and L have two atoms. Studied zero-divisor graph of a lattice L and N. Further, explored relation between both zero divisor graphs and they are complete bipartite graph.

Abstract Let M be a non-commutative gamma ring and Z Γ M ${Z}_{{\Gamma}}\left(M\right)$ denote the center of the gamma ring M . The vertices a and b are consecutive if a ≠ b and aαb = bαa for every α ∈ Γ, with vertices taken from the set M − Z Γ M $M-{Z}_{{\Gamma}}\left(M\right)$ . This graph is called the commuting graph of the gamma ring M . We show that the complement graph of the commuting graph of M is connected but not a complete bipartite graph. Additionally, if the diameter of the complement graph of the commuting graph is 1, then M $\left\vert M\right\vert $ must equal 4. It is also shown that the complement graphs of the commuting graphs for all Γ-rings of order greater than 16 are planar. Furthermore, the commuting graph of a Γ-ring M , where the order is p 2 or p 3 for a prime p , is the disjoint union of some complete graphs.