In this paper, we introduce a new concept of dimension on fractal networks, the so-called Hausdorff Steiner k-dimension. It is defined by a natural generalization of the Hausdorff dimension in Zeng's work. In particular, this new dimension is a non-trivial higher estimate for the Hausdorff dimension. We explore its basic properties and determine the exact values of some networks. Meanwhile, we discuss the changes under different graph operations, including single node sum and the Cartesian product of graphs. Within a specific class of networks, the Hausdorff Steiner k-dimension and the Hausdorff dimension exhibit an exact quantitative relationship.

Abstract In this paper, we study the existence of solutions for equilibrium problems in which the bifunction involved is defined on the Cartesian product of two distinct sets. Such a problem will be called a generalized equilibrium problem. The proofs we propose for establishing the existence theorems do not rely on classical tools, such as KKM-type theorems or separation theorems. Instead, as we will see, they are based on an intersection theorem established by Bassanezi and Greco [11]. Upper semicontinuity in the first variable of a given bifunction is a standard condition when seeking solutions to a classical equilibrium problem. In this paper, we introduce a concept that refines upper semicontinuity, called marginal r -upper semicontinuity. In our existence results, it will suffice as the bifunction to be marginally 0-upper semicontinuous in the first variable.

We defined Krull modules over an integral domain as a completely integrally closed module satisfying ascending chain condition on integral v-submodules. However, if the dimension of the module of fractions over the quotient field is more than 1, then there are many non-integral submodules. Thus, we then defined strongly Krull modules which involves all non-zero submodules (whether it’s integral or non-integral). In this paper, we will prove that the Cartesian product of the ring of integers and the ring of p-integers is an example of Krull module which is not strongly Krull, showing that our definition of strongly Krull modules is indeed properly stronger than Krull modules.

UDC 519.17 The forgotten topological index denoted by $F(G)$ of a graph $G=(V,E)$ is defined as follows: $F(G)=\displaystyle\sum\nolimits_{i=1}^{n}\!d_v^3,$ where $d_v$ denotes the degree of the vertex $v.$ We extend the notion of forgotten topological index to signed graphs and introduce the $MS$-index of a signed graph. Moreover, we determine the forgotten topological index for the tensor product, Cartesian product, lexicographic product, strong product, symmetric difference, and the joint of the graphs $G_1$ and $G_2$ in terms of the forgotten topological index, the first Zagreb index, and the $M\kern-1ptS$-index of signed graphs $\Sigma_1=(G_1,\sigma_1)$ and $\Sigma_2=(G_2,\sigma_2),$ along with their sizes and orders.

The eccentricity matrix 𝔼(G) of a graph G is constructed from the distance matrix D(G) by only retaining the leading values in each row and each column and setting the rest of the elements in the corresponding row and column to zero. The article aims to study the concepts of energy and the inertia of the eccentricity matrices of graphs obtained by means of various graph products, like the strong product, the Cartesian product, and the Kronecker product, and well-established relations are derived in terms of their base graphs. Furthermore, we construct infinitely many pairs of non-isomorphic graphs having the same set of eigenvalues with respect to the eccentricity matrix.

We introduce the concept of plithogenic signed graphs and define several binary operations on them, including union, join and Cartesian product to model the combination of heterogeneous networks. These operations provide a flexible mathematical framework for integrating multi-attribute and polarity-based relationships. The proposed model is applied to analyse interaction dynamics in complex systems. In particular, it indicates that the absence of appropriate precautionary measures may lead to a higher risk of severe outbreaks in the future.

Soft set theory provides a flexible framework for modeling uncertainty, and its application to graph theory has led to the development of soft graphs and their various operations. In this paper, we explore the relationships among different soft graph products from a soft subgraph perspective. Specifically, we establish and prove soft subgraph relations among major soft graph products, including Cartesian, lexicographic, co-normal, strong, tensor, modular, and rooted products. The study shows that the Cartesian product of two soft graphs is a soft subgraph of the lexicographic, co-normal, and strong products; the lexicographic product is a soft subgraph of the co-normal product; the tensor product is a soft subgraph of the lexicographic, modular, normal, and co-normal products; the strong product is a soft subgraph of the lexicographic and co-normal products; and the rooted product is a soft subgraph of the Cartesian, lexicographic, co-normal, and strong products. Illustrative examples are provided to validate these theoretical results. The findings contribute to a clearer structural understanding of interrelations among soft graph products.

Abstract For , let be the self‐similar set in generated by the iterated function system . In this paper, we investigate the intersection of the unit circle with the Cartesian product . We prove that for , the intersection is trivial , that is, If , then the intersection is nontrivial. In particular, if , the intersection is of cardinality continuum. Furthermore, the bound is sharp: there exists a sequence with such that is nontrivial for all . This result provides a negative answer to a problem posed by Yu (2023). Our methods extend beyond the unit circle and remain effective for many nonlinear curves. We also characterize the intersection of missing digits Cantor sets with the sequence by utilizing the Legendre symbol.

Moving through Cartesian products, coronas and joins in general position

Erratum to “The thickness of amalgamations and Cartesian product of graphs" [Discuss. Math. Graph Theory 37(2017) 561–572

This research explores the Edge Hyper-Zagreb index, a unique method of calculating graph indices by replacing vertex degrees with edge degrees. The edge degree is obtained by summing the degrees of the end vertices and subtracting 2. We analyze the Edge Hyper- Zagreb index for the Cartesian product and join of graphs and compute it for linear Acenes molecules with the formula (C4n+2 H2n+4). Besides, we highlight the correlation between topological indices and the quantity of rings, using the Topological Indices Method (TIM) to predict the diverse thermodynamic characteristics of the Acenes family, like electron affinity, bond strength, heat of formation, and gap energy

Let T be a graph obtained by taking the Cartesian product of n cycles of even lengths ≥ 4 . It is known that for any subset of k vertices of T with 1 ≤ k ≤ 2 n − 1 , there exists a 2-factor in T where each cycle contains exactly one of the k vertices. We generalize this result to r-factors by proving that for 1 ≤ k ≤ 2 n − r + 1 and 2 ≤ r ≤ 2 n , there exists an r-factor H in T such that each component of H is an r-regular, r-connected and bipancyclic graph, with each component containing exactly one of the k specified vertices of T. As a consequence, we obtain an analogous result for hypercubes.

Abstract We introduce a relaxation of stability, called almost sure stability, which is insensitive to perturbations by subsets of Loeb measure $0$ in a non-standard finite group. We show that almost sure stability satisfies a stationarity principle in the sense of geometric stability theory for measure-independent elements. We apply this principle to deduce the existence of squares in dense almost surely stable subsets of Cartesian products of non-standard finite groups, possibly non-abelian. Our results imply qualitative asymptotic versions for Cartesian products of finite groups. In the final section, we establish the existence of $3\times 2$ -grids (and thus of L -shapes) in dense almost surely stable two-dimensional subsets of finite abelian groups of odd order.

Summary Spatial variables can be observed in many different forms, such as regularly sampled random fields (lattice data), point processes and randomly sampled spatial processes. Joint analysis of such collections of observations is clearly desirable, but complicated by the lack of an easily implementable analysis framework. We fill this gap by providing a multitaper analysis framework using coupled discrete and continuous data tapers, combined with the discrete Fourier transform for inference. Using this set of tools is important, as it forms the backbone for practical spectral analysis. In higher dimensions it is especially important not to be constrained to Cartesian product domains, and so we develop the methodology for spectral analysis using irregular domain data tapers and the tapered discrete Fourier transform. We discuss its fast implementation, as well as the asymptotic and large finite-domain properties. Estimators of partial association between different spatial processes are provided, as are principled methods to determine their significance, and we demonstrate their practical utility using a large-scale ecological dataset.

Abstract A path factor in a graph G is a factor of G in which every component is a path on at least two vertices. Let $$T\Box P_n$$ T □ P n be the Cartesian product of a tree T and a path on n vertices. Kao and Weng [11] proved that $$T\Box P_n$$ T □ P n is hamiltonian if T has a path factor, n is an even integer and $$n\ge 4\Delta (T)-2$$ n ≥ 4 Δ ( T ) - 2 . They conjectured that for every $$\Delta \ge 3$$ Δ ≥ 3 there exists a connected graph G of maximum degree $$\Delta $$ Δ which has a path factor, such that for every even $$n< 4\Delta -2$$ n < 4 Δ - 2 the product $$G\Box P_n$$ G □ P n is not hamiltonian. In this article we prove this conjecture.

The concept of edge connectivity was first proposed by K. Menger, and in communication networks and logical networks, edge connectivity can be used to measure network reliability and fault tolerance. The graph product method can be used to construct complex networks, simulate biological molecule interactions etc. At present, research on the edge connectivity of product graphs mainly focuses on the connectivity of standard product graphs, such as Cartesian product graphs, strong product graphs. The unique properties exhibited by non standard product graphs (such as semi-strong product graphs.) in practical applications are worth further exploration. The concept of semi-strong product was proposed by Mordeson and Chang Shyh, that is, for two graphs and , their semi-strong product is a graph whose vertex set is , and the edge set is defined as follows: if and are two vertices in the semi-strong product , then there is an edge between them if and only if and and are adjacent in , or and are adjacent in and and are adjacent in . And applications of the semi-strong product in fuzzy graphs, symbolic graphs, and finance have shown its broad research prospects. In this article, we mainly study the edge connectivity of semi-strong product graphs, and obtain some exact values. Furthermore, we also give an necessary and sufficient condition for a semi-strong product to be maximally edge-connected.

Abstract This paper investigates the approximation of functions with finite smoothness defined on domains with a Cartesian product structure. The recently proposed tensor product multilevel method (TPML) combines Smolyak’s sparse grid method with a kernel-based residual correction technique. The contributions of this paper are twofold. First, we present two improvements on the TPML that reduce the computational cost of point evaluations compared to a naive implementation. Second, we provide numerical examples that demonstrate the effectiveness and innovation of the TPML.

A complex network of processors and communication channels, known as an interconnection network, is used by components in a parallel computing system to share data. In computer networks, where positioning various modules on the integrated circuit is one of the primary cost requirements, performing concurrent algorithms in these intricate, organized circuits requires the use of graph embedding. The placement issue in circuit designs, without any deterministic algorithms, can be resolved by determining the most suitable layout. Such a network allocation process can be carried out using graph embedding. This study establishes the optimal arrangement of processors to obtain minimum wirelength for embedding the Cartesian product of complete graphs into paths and grids.

In this article, we define the Cartesian, Categorical, and Lexicographic, and Strong products of quantum graphs. We provide bounds on the quantum chromatic number of these products in terms of the quantum chromatic number of the factors. To adequately describe bounds on the lexicographic product of quantum graphs, we provide a notion of a quantum b-fold chromatic number for quantum graphs.

The Imprimitivity Action of the Product of Finite Alternating Groups on Cartesian Product of Ordered γ- Tuples of Finite Sets