Cycles In Graphs Research Articles

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

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

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.

Paired coalition in graphs

A paired coalition in a graph G = ( V , E ) consists of two disjoint sets of vertices V 1 and V 2 , neither of which is a paired dominating set but whose union V 1 ∪ V 2 is a paired dominating set. A paired coalition partition (abbreviated pc-partition) in a graph G is a vertex partition π = { V 1 , V 2 , … , V k } such that each set V i of π is not a paired dominating set but forms a paired coalition with another set V j ∈ π . The paired coalition graph PCG ( G , π ) of the graph G with the pc-partition π of G, is the graph whose vertices correspond to the sets of π , and two vertices V i and V j are adjacent in PCG ( G , π ) if and only if their corresponding sets V i and V j form a paired coalition in G. In this paper, we initiate the study of paired coalition partitions and paired coalition graphs. In particular, we determine the paired coalition number of paths and cycles, obtain some results on paired coalition partitions in trees and characterize pair coalition graphs of paths, cycles and trees. We also characterize triangle-free graphs G of order n with PC ( G ) = n and unicyclic graphs G of order n with PC ( G ) = n − 2 .

Online Size Ramsey Numbers: Odd Cycles vs Connected Graphs

Online Size Ramsey Numbers: Odd Cycles vs Connected Graphs

Some Existence Results of Coupled Hilfer Fractional Differential System and Differential Inclusion on the Circular Graph

Some Existence Results of Coupled Hilfer Fractional Differential System and Differential Inclusion on the Circular Graph

Completely degenerate equilibria of the Kuramoto model on networks

Kuramoto Networks contain non-hyperbolic equilibria whose stability is sometimes difficult to determine. We consider the extreme case in which all Jacobian eigenvalues are zero. In this case linearizing the system at the equilibrium leads to a Jacobian matrix which is zero in every entry. We call these equilibria completely degenerate. We prove that they exist for certain intrinsic frequencies if and only if the underlying graph is bipartite, and that they do not exist for generic intrinsic frequencies. In the case of zero intrinsic frequencies, we prove that they exist if and only if the graph has an Euler circuit such that the number of steps between any two visits at the same vertex is a multiple of 4. The simplest example is the cycle graph with 4 vertices. We prove that graphs with this property exist for every number of vertices N⩾6 and that they become asymptotically rare for N large. Regarding stability, we prove that for any choice of intrinsic frequencies, any coupling strength and any graph with at least one edge, completely degenerate equilibria are not Lyapunov stable. As a corollary, we obtain that stable equilibria in Kuramoto Networks must have at least one strictly negative eigenvalue.

Edge Metric Dimension of Comb Product Cycles Versus Stars and Fans

The edge metric dimension finding of a graph resulting from the comb product on a cycle graph to the complete graph and the fact about the dominant vertex in the complete graph arise a problem about the edge metric dimension of a graph resulting from the comb product on a cycle graph to the other graphs that have dominant vertex such as star and fan graph. This study aims to determine the edge metric dimension of the graph resulting from the comb product on a cycle graph to the star and fan graph, respectively. The results show that the upper and lower bound of the edge metric dimension of both graphs are equivalent so that an exact edge metric dimension is obtained. The edge metric dimension of the graph comb product of the cycle graph C_n to the star graph S_(1,m) is n(m-1). The edge metric dimension of the graph comb product of the cycle graph C_n to the fan graph F_(1,m) is n(m-1).

On the <i>k</i>-restricted Intersection Graph

The problem on intersection graph was introduced by Szpilrajn-Marczewski in 1945. This study introduces a new variant of intersection graph, called the $k$-restricted intersection graph. Let $\mathcal{S}_n$ be a nonempty $n$-element set, for some positive integer $n$ and let $S_{(n,k)}$ be the set of all the $k$-element subsets of $\mathcal{S}_n$ where $0\leq k\leq n$. A $k$-restricted intersection graph, denoted by $G_{S_{(n,k)}}$, is a graph with vertex set $S_{(n,k)}$ such that two vertices $A,B\in S_{(n,k)}$ are adjacent whenever $A\cap B\neq \emptyset$ and $A\neq B$. Here, we determined the order and size of $G_{S_{(n,k)}}$. Moreover, some parameters such as independence number, domination number, and isolate domination number of the $k$-restricted intersection graph were established. Finally, necessary and sufficient conditions for a $G_{S_{(n,k)}}$ to be isomorphic to the cycle graph and complete graph were determined.

Graceful Antimagic Path and Cycle Related Graphs

A graph with vertices and edges is said to be antimagic if its edges can be labeled with such that the weights of vertices of are pairwise distinct. The graceful labeling of a graph with edges is an assignment of integers from the set to the vertices of , such that no two vertices receive the same label, where each edge is assigned the absolute value of the difference between the labels of its end vertices and the resulting edge labeling runs from to is inclusive. Moreover, if the induced edge labeling is simultaneously antimagic, that is, the sums of the labels of all edges incident to a given vertex are pairwise distinct for different vertices, we call the graceful labeling graceful antimagic. In this study, we will exhibit the existence of graceful antimagic labeling for two families of graphs the first derived from the path graph Pn, and the second from the cycle graph Cn. Both families were derived using the idea of a rooted product between the two graphs.