graph-theory

IntroductionWith the deep integration of generation-transmission-load-storage systems, the power demand side has become highly dynamic and stochastic, challenging the traditional assumption that user behavior remains stationary over time. Static clustering models therefore suffer from sensitivity to daily noise and false user identity switching.MethodsThis study proposes Dynamic Evolutionary Clus…

A finite, undirected graph $G=(V,E)$ is connected in the traditional sense if for all $v, w\in V$ with $v\neq w$ there is a finite path from $v$ to $w$. Moreover, $G$ is connected in the topological ...

Fix integers and and set . Let denote the complete -partite -uniform hypergraph with parts of size . We prove that the Zarankiewicz number provided . Previously this was known only for due to Pohoata and Zakharov. Our novel approach, which uses Behrend’s construction of sets with no 3-term arithmetic progression, also applies for small values of , for example, it gives where the exponent 11/4 is …

We call a hypergraph $H=(V,E)$ bipartite if there is $S\subseteq V$ with $e\cap S$ and $e\setminus S$ both nonempty for all $e\in E$. If $(\omega, E)$ is not bipartite and all members are infinite, do ...

Probability can become hard to reason about when many variables interact. One variable affects another. Evidence changes belief. Dependencies start to form a network. That is where Probabilistic Graphical Models become useful. Core Idea A Probabilistic Graphical Model represents uncertainty with a graph. The nodes are random variables. The edges represent relationships between them. Instead of tr…

Let [Formula: see text] be a group. A weak Cayley table isomorphism is a bijection [Formula: see text] such that (i) [Formula: see text] is conjugate to [Formula: see text] for [Formula: see text], [Formula: see text] and (ii) [Formula: see text] sends conjugacy classes to conjugacy classes. The set of all such bijections forms a group [Formula: see text]. We study [Formula: see text] for 56 of t…

For a nontrivial [Formula: see text]-group [Formula: see text], a maximal chain of [Formula: see text] is a chain of subgroups [Formula: see text] with [Formula: see text] for [Formula: see text], where [Formula: see text] is an integer. We determine the structure of finite groups having [Formula: see text]-quasinormality of a maximal chain of Sylow subgroups. We obtain new characterizations of f…

Let [Formula: see text] be a finite group and [Formula: see text]. Then [Formula: see text] is called [Formula: see text]-decomposable in [Formula: see text] if it is a union of [Formula: see text] distinct [Formula: see text]-conjugacy classes, and denote [Formula: see text] by [Formula: see text]. A group [Formula: see text] is called [Formula: see text]-decomposable with [Formula: see text]. T…

A graph is edge-transitive if its automorphism group acts transitively on the set of edges of the graph. In this paper, we classify edge-transitive 8-valent graphs of order [Formula: see text] for each prime [Formula: see text].

Abstract Let 𝐺 be a permutation group, with minimal degree <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>μ</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>G</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> \mu(G) and base size <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>b</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo…

The Royal Hybrid Number 1

This document develops a strict graph-theoretic formalization of the determination dependency graph G_D introduced in natural language in Structural Determination Theory (SDT) v5.9. Working at the level of partial-order theory, lattice theory, and elementary category theory, the formalization elevates SDT's core structural objects — nodes, edges, products, layers, traces — to a precisely defined …

I am taking a graph theory class and our professor started a new topic called graph labelling today, specifically, vertex labelling. Halfway through the lecture, I dozed off and missed the rest of it. ...

A subgroup [Formula: see text] of a group [Formula: see text] is called a TI-subgroup of [Formula: see text] if [Formula: see text] or [Formula: see text] for each [Formula: see text] [Formula: see text][Formula: see text]. Furthermore, let [Formula: see text] be a partition of the set of all primes [Formula: see text], a subgroup [Formula: see text] of [Formula: see text] is called [Formula: see…

Bayesian Networks can feel confusing because they combine two things at once. Graphs show structure. Probabilities show uncertainty. The key is to see them as one model, not two separate topics. Core Idea A Bayesian Network represents relationships between variables using a directed graph. Each node is a variable. Each edge shows a dependency. Each node also has probability values that explain ho…

Let Γ be a finite group and S = S−1 a non-central conjugacy class generating Γ. The full-context relation module on the Cayley digraph Cay(Γ, S) generates, by orthogonal complement and wedge-component projection inside the σ-anti-invariant ambient, a finite-dimensional Γ-module under based conjugation: the post-vanishing Cayley carrier E(2),πx,full(Cay(Γ, S))−. This paper studies its representati…

In this paper, we determine the edge metric dimension of the comb product of a cycle graph and a simple graph containing a dominant vertex. This result generalizes previous findings on the edge metric dimension of the comb product of a cycle and a complete graph. We show that the edge metric dimension of C n ▷ D , where D is a simple graph with a dominant vertex, equals the product of the order o…

The matching book embedding of a graph G is an embedding of G with the vertices on the spine, and each edge within a single page so that the edges on each page do not intersect and the degree of vertices on each page is at most one. The matching book thickness of G is the minimum number of pages in a matching book embedding of G, denoted by mbt(G). In this paper, the exact matching book thickness…

The capability of one architecture to simulate another serves as the foundation for network comparison, with embedding playing a key role in analyzing these simulations. In architectural simulation, graph embedding is one of the most powerful techniques for executing parallel algorithms and modeling diverse interconnection networks. In our earlier work, we listed an open problem that the determin…