Cycles In Graphs Research Articles

Simple Recurrent Networks are Interactive.

There is disagreement among cognitive scientists as to whether a key computational framework - the Simple Recurrent Network (SRN; Elman, Machine Learning, 7(2), 195-225, 1991; Elman, Cognitive Science, 14(2), 179-211, 1990) - is a feedforward system. SRNs have been essential tools in advancing theories of learning, development, and processing in cognitive science for more than three decades. If SRNs were feedforward systems, there would be pervasive theoretical implications: Anything an SRN can do would therefore be explainable without interaction (feedback). However, despite claims that SRNs (and by extension recurrent neural networks more generally) are feedforward (Norris, 1993), this is not the case. Feedforward networks by definition are acyclic graphs - they contain no loops. SRNs contain loops - from hidden units back to hidden units with a time delay - and are therefore cyclic graphs. As we demonstrate, they are interactive in the sense normally implied for networks with feedback connections between layers: In an SRN, bottom-up inputs are inextricably mixed with previous model-internal computations. Inputs are transmitted to hidden units by multiplying them by input-to-hidden weights. However, hidden units simultaneously receive their own previous activations as input via hidden-to-hidden connections with a one-step time delay (typically via context units). These are added to the input-to-hidden values, and the sums are transformed by an activation function. Thus, bottom-up inputs are mixed with the products of potentially many preceding transformations of inputs and model-internal states. We discuss theoretical implications through a key example from psycholinguistics where the status of SRNs as feedforward or interactive has crucial ramifications.

The maximum number of odd cycles in a planar graph

AbstractHow many copies of a fixed odd cycle, , can a planar graph contain? We answer this question asymptotically for and prove a bound which is tight up to a factor of 3/2 for all other values of . This extends the prior results of Cox and Martin and of Lv, Győri, He, Salia, Tompkins, and Zhu on the analogous question for even cycles. Our bounds result from a reduction to the following maximum likelihood question: which probability mass on the edges of some clique maximizes the probability that edges sampled independently from form either a cycle or a path?

XNLP-Completeness for Parameterized Problems on Graphs with a Linear Structure

AbstractIn this paper, we showcase the class XNLP as a natural place for many hard problems parameterized by linear width measures. This strengthens existing W[1]-hardness proofs for these problems, since XNLP-hardness implies W[t]-hardness for all t. It also indicates, via a conjecture by Pilipczuk and Wrochna (ACM Trans Comput Theory 9:1–36, 2018), that any XP algorithm for such problems is likely to require XP space. In particular, we show XNLP-completeness for natural problems parameterized by pathwidth, linear clique-width, and linear mim-width. The problems we consider are Independent Set, Dominating Set, Odd Cycle Transversal, (q-)Coloring, Max Cut, Maximum Regular Induced Subgraph, Feedback Vertex Set, Capacitated (Red-Blue) Dominating Set, Capacitated Vertex Cover and Bipartite Bandwidth.

A framework for generalizing toric inequalities for holographic entanglement entropy

We conjecture a multi-parameter generalization of the toric inequalities of [1]. We then extend their proof methods for the generalized toric inequalities in two ways. The first extension constructs the graph corresponding to the toric inequalities and the generalized toric conjectures by tiling the Euclidean space. An entanglement wedge nesting relation then determines the geometric structure of the tiles. In the second extension, we exploit the cyclic nature of the inequalities and conjectures to construct cycle graphs. Then, the graph can be obtained using graph Cartesian products of cycle graphs. In addition, we define a set of knots on the graph by following [1]. These graphs with knots then imply the validity of their associated inequality. We study the case where the graph can be decomposed into disjoint unions of torii. Under the specific case, we explore and prove the conjectures for some ranges of parameters. We also discuss ways to explore the conjectured inequalities whose corresponding geometries are d-dimensional torii (d > 2).

Zagreb Polynomials, Co-Polynomials of Bipartite and Strongly Bipartite Graphs

First Zagreb polynomial of a graph G with vertex set V(G) and edge set E(G) is defined as M1(G,x) =∑uv∈E(G)xdu+dv and the first Zagreb index can be obtained from it as M1(G) = M1(G,x)|x=1. In this paper Zagreb polynomials, co-polynomials and corresponding Zagreb indices, co-indices are obtained for path, cycle, double and strong double graphs.

A new virtual water qualimeter with a changeable broken linear pointer based on a quadrant circular constellation graph

ABSTRACT As water is a very important natural resource for humans and its quality is closely related to human health, water quality evaluation (WQE) is very important for the protection of human health, livelihoods, environment, and management of water. Although various WQE methods have been developed and used, a common drawback is that the methods lack visualization of the WQE results and require additional complicated graphical techniques. The Pointer meter is an analogue measuring device with a pointer indicating the measured value with its location. A constellation graph (CG) enables to map multi-dimensional data on a two-dimensional plane. By introducing the pointer meter and CG for WQE, this paper proposes a new visualized pointer meter-type virtual water qualimeter (VWQ-meter) based on a quadrant circular constellation graph (QCCG). It has a changeable broken linear pointer with variable lengths and shapes, while the traditional pointer meters have straight-line pointers with a fixed length and a straight-line shape. It evaluates the water quality not only quantitatively but also intuitively as in the pointer meters. To demonstrate its effectiveness, it is applied to one example of WQE. It could be widely applied to not only WQE but also various multi-criteria decision-making (MCDM) problems arising in water pollution evaluation, water resource management, water supply, and wastewater treatment.

Cyclic orders and graphs of groups

Abstract We examine a cyclic order on the directed edges of a tree whose vertices have cyclically ordered links. We use it to show that a graph of groups with left-cyclically ordered vertex groups and convex left-ordered edge groups is left-cyclically orderable.

Exploring Mean Cordial Labeling Possible Combination Position of Vertices in Graphs: A Computational Approach

Objectives: We have developed the Python module of mean cordial labeling for the different types of graphs, like Path, Cycle, and Subdivision of Star graphs. We aim to identify the number of combination positions of vertices that satisfy the mean cordial labeling rules. We have found mathematical equations that describe the labeling behavior in these graphs. Methods: In this paper, a Python program was developed to find the mean cordial labeling of the Path, Cycle, and Subdivision of the Star graph. We have obtained the number of combination positions of vertex obeying the mean cordial labeling condition using this Python module. We have drawn the graph between the number of vertices and the number of combination positions of the vertex. Findings: We have found that the different graph types like Path, Cycle, and Subdivision of Star graph satisfy the mean cordial labeling condition. The result indicates that the number of combination positions of the Path and cycle graph fulfill the power equation Y=a*xb and the Subdivision of the Star graph meets the exponential growth equation Y=a*bx. Novelty: In this paper, a new method has been developed to find the number of combination position of varies graphs using the Python module. This computational approach will be useful to analyze the biological networking and protein-protein interaction study. Keywords: Mean Cordial, Path, Cycle, Subdivision of Star graph, Computational graph theory

Spectral Extremal Problem on $t$ Copies of $\ell$-Cycles

Extremal problem on cycles plays an important role in extremal graph theory. Let $ex(n,F)$ and $spex(n,F)$ be the maximum size and spectral radius over all $n$-vertex $F$-free graphs, respectively. In this paper, we shall pay attention to the study of both $ex(n,tC_\ell)$ and $spex(n,tC_\ell)$. On the one hand, we determine $ex(n,tC_{2\ell+1})$ and characterize the extremal graph for any integers $t,\ell$ and $n\geq f(t,\ell)$, where $f(t,\ell)=O(t\ell^2)$. This generalizes the result on $ex(n,tC_3)$ of Erdős [Arch. Math. 13 (1962) 222–227] as well as the research on $ex(n,C_{2\ell+1})$ of Füredi and Gunderson [Combin. Probab. Comput. 24 (2015) 641–645]. On the other hand, motivated by the spectral Turán-type problem proposed by Nikiforov, we obtain the extremal spectral radius $spex(n,tC_{\ell})$ for any fixed $t,\ell$ and large enough $n$. Our results extend some classic spectral extremal results or conjectures on odd cycles and even cycles. Our results also give some inspirations for general spectral Turán-type problem $spex(n,F)$ on bipartite or non-partite $F$.

Exact and parameterized algorithms for the independent cutset problem

The Independent Cutset problem asks whether there is a set of vertices in a given graph that is both independent and a cutset. Such a problem is ▪-complete even when the input graph is planar and has maximum degree five. In this paper, we first present a O⁎(1.4423n)-time algorithm to compute a minimum independent cutset (if any). Since the property of having an independent cutset is MSO1-expressible, our main results are concerned with structural parameterizations for the problem considering parameters incomparable with clique-width. We present ▪-time algorithms for the problem considering the following parameters: the dual of the maximum degree, the dual of the solution size, the size of a dominating set (where a dominating set is given as an additional input), the size of an odd cycle transversal, the distance to chordal graphs, and the distance to P5-free graphs. We close by introducing the notion of α-domination, which allows us to identify more fixed-parameter tractable and polynomial-time solvable cases.

On monoids of endomorphisms of a cycle graph

Abstract In this paper, we consider endomorphisms of an undirected cycle graph from Semigroup Theory perspective. Our main aim is to present a process to determine sets of generators with minimal cardinality for the monoids wEndCn and EndCn of all weak endomorphisms and all endomorphisms of an undirected cycle graph Cn with n vertices. We also describe Green’s relations and regularity of these monoids and calculate their cardinalities.

On the Constructor–Blocker game

AbstractIn the Constructor–Blocker game, two players, Constructor and Blocker, alternately claim unclaimed edges of the complete graph . For given graphs and , Constructor can only claim edges that leave her graph ‐free, while Blocker has no restrictions. Constructor's goal is to build as many copies of as she can, while Blocker attempts to minimize the number of copies of in Constructor's graph. The game ends once there are no more edges that Constructor can claim. The score of the game is the number of copies of in Constructor's graph at the end of the game when both players play optimally and Constructor plays first. In this paper, we extend results of Patkós, Stojaković and Vizer on to many pairs of and : We determine when and , also when both and are odd cycles, using Szemerédi's Regularity Lemma. We also obtain bounds of when and .

Counting circuit double covers

AbstractWe study a counting version of Cycle Double Cover Conjecture. We discuss why it is more interesting to count circuits (i.e., graphs isomorphic to for some ) instead of cycles (graphs with all degrees even). We give an almost‐exponential lower bound for graphs with a surface embedding of representativity at least 4. We also prove an exponential lower bound for planar graphs. We conjecture that any bridgeless cubic graph has at least circuit double covers and we show an infinite class of graphs for which this bound is tight.

Absolute Zeta functions and periodicity of quantum walks on cycles

The quantum walk is a quantum counterpart of the classical random walk. On the other hand, absolute zeta functions can be considered as zeta functions over $\mathbf{F}_1$. This study presents a connection between quantum walks and absolute zeta functions. In this paper, we focus on Hadamard walks and $3$-state Grover walks on cycle graphs. The Hadamard walks and the Grover walks are typical models of the quantum walks. We consider the periods and zeta functions of such quantum walks. Moreover, we derive the explicit forms of the absolute zeta functions of corresponding zeta functions. Also, it is shown that our zeta functions of quantum walks are absolute automorphic forms.

Computing Zagreb Connection Indices for the Cartesian Product of Path and Cycle Graphs

Topological indices ( ) are a class of graph-based descriptors widely used in chemiformatics and Quantitative Structure-Activity Relationship (QSAR) studies. TIs capture key structural features of molecules by encoding graph-theoretic information, offering a quantitative representation of molecular topology. Each mathematical network is associated with a specific numerical value determined by a function. Among the Zagreb connection indices have been extensively researched. This study delves into the Cartesian product of cycle graphs with path graphs, elucidating the comprehensive implications of . These indices encompass the first , second , and third . Furthermore, comprehensive results of the modified first , modified second , and modified third are presented, along with the first multiplicative ZCI second multiplicative ZCI (SMZCI), and third multiplicative ZCI Moreover, modified first multiplicative ZCI ), modified second multiplicative ZCI and modified third multiplicative ZCI ( are also calculated. To provide precision, both the graphical and numerical analyses of the computed findings are aligned for the two Cartesian products. The foundation for mathematically modeling complex networks and chemical structures lies in graph theory. Topological indices ( ) are a class of graph-based descriptors widely used in chemiformatics and quantitative structure-activity relationship (QSAR) studies. TIs capture key structural features of molecules by encoding graph-theoretic information, offering a quantitative representation of molecular topology. Each mathematical network is associated with a specific numerical value determined by a topological index function. Among these the Zagreb connection indices are extensively researched TIs. This article delves into the Cartesian product of cycle graphs with path graphs, elucidating the comprehensive implications of . These indices encompass the first, Second and third. Furthermore, we present the comprehensive results of the modified , modified and modified , and also present the first multiplicative Zagreb connection index , and , along with their modified counterparts like modified first multiplicative Zagreb connection index , and . These analysis encompass two distinct types of graphs: cycle graphs and path graphs. To provide further precision, we align both the graphical and numerical analysis of our computed findings for these two Cartesian products.

Further results on enumeration of perfect matchings of Cartesian product graphs

Abstract Counting perfect matchings is an interesting and challenging combinatorial task. It has important applications in statistical physics and chemistry. As the general problem is #P-complete, it is usually tackled by randomized heuristics and approximation schemes. Let G G and H H be two graphs. Denote by G × H G\times H the Cartesian product of graphs G G and H H . The number of perfect matchings of some types of Cartesian product G × H G\times H is investigated by the Pfaffian method, where G ≅ P 2 G\cong {P}_{2} , P 3 {P}_{3} , P 4 {P}_{4} or C 4 {C}_{4} , and H ≅ T H\cong T or H H is a non-bipartite graph with a unique cycle. In this article, we construct a Pfaffian orientation of the graph G × P 2 G\times {P}_{2} , where G G is a non-bipartite graph with k k odd cycles. We obtain the counting formula on the number of perfect matchings of G × P 2 G\times {P}_{2} .

Depth and Stanley depth of powers of the path ideal of a cycle graph

Depth and Stanley depth of powers of the path ideal of a cycle graph

The h-vectors of edge rings of odd cycle compositions

The h-vectors of edge rings of odd cycle compositions

On the Generalized Turán Problem for Odd Cycles

On the Generalized Turán Problem for Odd Cycles

Quantum-assisted rendezvous on graphs: explicit algorithms and quantum computer simulations

Abstract We study quantum advantage in one-step rendezvous games on simple graphs analytically, numerically, and using noisy intermediate-scale quantum (NISQ) processors. Our protocols realise the recently discovered (Mironowicz 2023 New J. Phys. 25 013023) optimal bounds for small cycle graphs and cubic graphs. In the case of cycle graphs, we generalise the protocols to arbitrary graph size. The NISQ processor experiments realise the expected quantum advantage with high accuracy for rendezvous on the complete graph K 3. In contrast, for the graph 2 K 4 , formed by two disconnected 4-vertex complete graphs, the performance of the NISQ hardware is sub-classical, consistent with the deeper circuit and known qubit decoherence and gate error rates.