Cycles In Graphs Research Articles

On star-k-PCGs: exploring class boundaries for small k values

A graph G=(V,E) is a star-k-pairwise compatibility graph (star-k-PCG) if there exists a weight function w:V→R+ and k mutually exclusive intervals I1,I2,…Ik, such that there is an edge uv∈E if and only if w(u)+w(v)∈⋃iIi. These graphs are related to two important classes of graphs: pairwise compatibility graphs (PCGs) and multithreshold graphs. It is known that for any graph G there exists a k such that G is a star-k-PCG. Thus, for a given graph G it is interesting to know which is the minimum k such that G is a star-k-PCG. We define this minimum k as the star number of the graph, denoted by γ(G). Here we investigate the star number of simple graph classes, such as graphs of small size, caterpillars, cycles and grids. Specifically, we determine the exact value of γ(G) for all the graphs with at most 7 vertices. By doing so we show that the smallest graphs with star number 2 are only 4 and have exactly 5 vertices; the smallest graphs with star number 3 are only 3 and have exactly 7 vertices. Next, we provide a construction showing that the star number of caterpillars is one. Moreover, we show that the star number of cycles and two-dimensional grid graphs is 2 and that the star number of 4-dimensional grids is at least 3. Finally, we conclude with numerous open problems.

Odd Cycles in Conditionally Faulty Enhanced Hypercube Networks

The n-dimensional enhanced hypercube Qn,k(1≤k≤n−1) is a well-known variation of hypercube networks. Its structure can be obtained from the hypercube by adding 2n−1 complementary edges. We denote a network G to be a conditionally faulty model if every fault-free vertex of G connects at least two fault-free edges. Let Fv and Fe be the set of faulty vertices and faulty edges in Qn,k(1≤k≤n−1), respectively. In this paper, for the conditionally faulty Qn,k with |Fv|+|Fe|≤2n−5, where n(≥3) and k have different parity, I prove that Qn,k−Fv−Fe contains a fault-free cycle with every odd length l, where n−k+4≤l≤2n−2|Fv|−1.

Frequency of Occurrence of Solar Plasma Disturbances in Interplanetary Space

Abstract Major solar plasma disturbances are subjected to Lomb-Scargle periodogram and wavelet analysis to determine the occurrence frequency. These disruptions include interplanetary coronal mass ejection, sudden storm commencement, high-speed streams, corotating interaction regions, interplanetary shocks and Forbush decreases. We included information on all of the aforementioned solar disturbances for the last six solar cycles, from 1965 to 2023, for this study. Our findings reveal some intriguing and noteworthy results that clearly distinguish between even and odd-numbered solar cycles. The study suggests that the sun behaves differently in odd and even-numbered solar cycles as it comes from the massive solar eruptions. During even-numbered solar cycles, variations $\sim{44}-$day period is prominently observed in addition to solar rotation ($\sim{27}-$day) and extended solar ($\sim{36}-$day) rotation. However, in addition to solar rotation, prolonged solar rotation, and periods of around $44-$ days, we also detect a number of intermittent changes with nearly comparable amplitude during the odd-numbered solar cycles. The findings also demonstrate that, in contrast to odd-numbered solar cycles the emissions rate of these disruptions is more distinct and predictable during even-numbered solar cycles.

Regular graphs to induce even periodic Grover walks

Regular graphs to induce even periodic Grover walks

On Topologies on Simple Graphs and Their Applications in Radar Chart Methods

This paper introduces a novel topology (upper approximated G-topology) on vertex sets of graphs using rough upper approximation neighborhoods, extending prior work on graph-induced topologies. Key results include characterizing discrete/indiscrete topologies for complete graphs, cycle graphs, and bipartite graphs (Theorems 1–3). The discrete topology for cycle graphs Cn, n>5, is particularly insightful. Exploring further, we delve into the continuity and isomorphism of graph mappings. Subsequently, we apply these findings to enhance radar chart graphical methods through the analysis of corresponding graph structures. These applications demonstrate practical relevance, linking graph structures to data visualization.

Geodetic Number of Cyclic Graphs

Geodetic Number of Cyclic Graphs

Paw decompositions of diamond and some edge cycle graphs

Paw decompositions of diamond and some edge cycle graphs

Shellability of 3-cut complexes of squared cycle graphs

Shellability of 3-cut complexes of squared cycle graphs

Odd Vertex Magic Labeling of Cyclic and Path Graphs

The k_0, k_e are odd vertex magic constants of cyclic graphs when p is odd and even respectively. The odd vertex magic constants k_(p_0 ),k_(p_e ) are for path graphs when p is odd and even respectively.

A Novel Problem and Algorithm for Solving Cordial Labeling of Some Fifth Powers of Graphs

In this paper we introduce a novel application of cordial labeling using the fifth power of graphs, demonstrating its potential for understanding and studying specific graph structures. The resulting cordial labeling scheme for the fifth power of paths, cycles, fans, wheels, lemniscate and the union of fifth power of paths and cycles graphs can provide insights into the properties and structures of these graphs. It can be used to analyze its connectivity, symmetry, and other graph-theoretical characteristics.

Graph Theory and Molecular Behavior: Utilizing Platt Numbers and Polynomial Approximations for Boiling Point Estimations

ABSTRACTAnalyzing complex multivalued datasets, such as boiling points and molecular Platt numbers, poses significant challenges due to the intricate relationships within the data, which traditional methods struggle to capture effectively. To address this, the study introduces Edge Neighborhood Graphs Ne(G), a novel graph‐based approach where the neighborhoods of edges in a graph G = (V, E) are treated as vertices, and an edge is drawn between two vertices if their corresponding neighborhoods intersect. This method is explored using complete, cycle, and wheel graphs, with formulas derived for vertex degrees in Ne(G) to uncover structural patterns. Polynomial approximations are employed to apply Ne(G) to multivalued data, demonstrating its effectiveness in modeling complex relationships and providing accurate predictions. By offering a powerful new tool for analyzing intricate datasets, this study makes a unique contribution to graph theory and data modeling, with promising potential for application to other graph types and diverse datasets in the future.

Acyclic coloring of two operations of graphs

Abstract The coloring theory of graphs is a very important direction in graph theory. The graph coloring problem has a strong application background. Many practical problems such as timetabling problems, frequency allocation problems, traffic arrangements, circuit design, and storage problems can be transformed into graph coloring problems. A substantial amount of scholarly work has been dedicated to the exploration of acyclic coloring, resulting in a plethora of significant findings that significantly contribute to the theoretical framework of vertex coloring in graphs. An acyclic coloring of a graph G refers to a proper vertex coloring where the subgraph resulting from any pair of color classes does not encompass any cycles. We analyze acyclic coloring in splitting graphs and Cartesian products of paths and cycles by using the method of structural coloring, reductive proof, and mathematical induction. Moreover, we present the precise acyclic chromatic numbers for splitting graphs of cycles and Cartesian products of paths and cycles.

Turán problem of signed graph for negative odd cycle

Turán problem of signed graph for negative odd cycle

Adaptive Exoskeleton Device for Stress Reduction in the Ankle Joint Orthosis.

Treating ankle fractures in athletes, commonly resulting from training injuries, remains a significant challenge. Current approaches to managing both non-surgical and postoperative foot and ankle disorders have focused on integrating sensory systems into orthotic devices. Recent analyses have identified several gaps in rehabilitation strategies, especially regarding gait pattern reformation during recovery. This work aims to enhance rehabilitation effectiveness for patients with ankle injuries by controlling load distribution and monitoring joint flexion/extension angles, as well as the reactive forces during therapeutic exercises and walking. We developed an exoskeleton device model using SolidWorks 2024 software, based on data from two patients: one healthy and one with an ankle fracture. Pressure measurements in the posterior limb region were taken using the F-Socket system and a custom electromechanical sensor designed by the authors. The collected data were analyzed using the butterfly parameterization method. This research led to the development of an adaptive exoskeleton device that provided pressure distribution data, gait cycle graphs, and a diagram correlating foot angles with the duration of exoskeleton use. The device demonstrated improvement in the patients' conditions, facilitating a more normalized gait pattern. A reduction in the load applied to the ankle joint was also observed, with the butterfly parameter confirming the device's correct operation.

Integer programs with bounded subdeterminants and two nonzeros per row

We give a strongly polynomial-time algorithm for integer linear programs defined by integer coefficient matrices whose subdeterminants are bounded by a constant and that contain at most two nonzero entries in each row. The core of our approach is the first polynomial-time algorithm for the weighted stable set problem on graphs that do not contain more than k vertex-disjoint odd cycles, where k is any constant. Previously, polynomial-time algorithms were only known for k =0 (bipartite graphs) and for k =1. We observe that integer linear programs defined by coefficient matrices with bounded subdeterminants and two nonzeros per column can be also solved in strongly polynomial-time, using a reduction to b -matching.

THE LOCATING CHROMATIC NUMBER OF CHAIN(A,4,n) GRAPH

Let be a connected graph with a vertex coloringsuch that two adjacent vertices have different colors. We denote an ordered partition where is a color class with color-, consisting of vertices given color , for . The color code of a vertex in is a -vector: . where is the distance between a vertex in and for . If every two vertices and in have different color codes, , then is called the locating -coloring of . The minimum number of colors k needed in this coloring is defined as the locating chromatic number, denoted by . This paper determines the locating chromatic number of chain graph and the induction of two graphs . Graph is a cyclic graph , which is the identification of , for n>2.

The Chromatic Number of the Edge Corona Operation of Cycle Graph and Star Graph

One of the concepts in graph theory that can be analyzed is chromatic numbers of a graph and operation of two graphs. There are various kinds of operations of two graphs, one of which is the corona edge operation. This research aims to determine the chromatic number of the edge corona operation of graph Cn*K1,m and K1,m*Cn, where Cn is a cycle graph and K1,m is a star graph. The chromatic number is determined based on the pattern formed from several n and m values. The results of this research show that the chromatic number of the edge corona operation of graph Cn*K1,m is 4 for n= 3, 4, ... k and m=1, 2, 3, ..., l. The chromatic number of the edge corona operation of graph K1,m*Cn is 5 if n is odd number. and is 4 if n is even number.

Odd Cycle Transversal on P 5 -free Graphs in Polynomial Time

An independent set in a graph \(G\) is a set of pairwise non-adjacent vertices. A graph \(G\) is bipartite if its vertex set can be partitioned into two independent sets. In the Odd Cycle Transversal problem, the input is a graph \(G\) along with a weight function w associating a rational weight with each vertex, and the task is to find a minimum weight vertex subset \(S\) in \(G\) such that \(G-S\) is bipartite; the weight of \(S\) , \(\text{w}(S)=\sum_{v\in S}\text{w}(v)\) . We show that Odd Cycle Transversal is polynomial-time solvable on graphs excluding \(P_{5}\) (a path on five vertices) as an induced subgraph. The problem was previously known to be polynomial-time solvable on \(P_{4}\) -free graphs and NP -hard on \(P_{6}\) -free graphs [Dabrowski, Feghali, Johnson, Paesani, Paulusma and Rzążewski, Algorithmica 2020]. Bonamy, Dabrowski, Feghali, Johnson and Paulusma [Algorithmica 2019] posed the existence of a polynomial-time algorithm on \(P_{5}\) -free graphs as an open problem. This was later re-stated by Rzążewski [Dagstuhl Reports, 9(6): 2019], by Chudnovsky, King, Pilipczuk, Rzążewski, and Spirkl [SIDMA 2021] who gave an algorithm with running time \(n^{O(\sqrt{n})}\) for the problem, and by Agrawal, Lima, Lokshtanov, Saurabh, and Sharma [SODA 2024] who gave a quasi-polynomial time algorithm.

COMPUTING THE SECURE CONNECTED DOMINANT METRIC DIMENSION PROBLEM OF CLASSES OF GRAPHS

This paper investigates the NP-hard problem of finding the lowest secure connected domination metric dimension of graphs. If each vertex in can be uniquely recognized by its vector of distances to the vertices in Scddim, then every vertex set Scddim of a connected graph resolves . If the subgraph induced by Scddim is a nontrivial connected subgraph of , then the resolving set Scddim of is connected. That resolving set is dominating if each vertex in that is not an element of Scddim is a neighbor of some vertices in Scddim. If there is a in such that is a dominating set for any in , then the dominating set is secure. If for every , there exists such that is a resolving set, then the resolving set is secure. These four cardinality values are the metric dimension of , the connected metric dimension of , the secure metric dimension of , and the connected domination metric dimension of , respectively. They correspond to the cardinality of the smallest resolving set of , the minimal connected resolving set, the minimal secure resolving set, and the minimal connected domination resolving set. In this paper, we introduce the secure connected domination metric dimension of graphs. If each vertex in G can be uniquely recognized by its vector of distances to the vertices in Scddim, then every vertex set Scddim of a connected graph resolves G. If the subgraph induced by Scddim is a nontrivial connected subgraph of G, then the resolving set Scddim of G is connected. That resolving set is dominating if each vertex in G that is not an element of Scddim is a neighbor of some vertices in Scddim. If there is a v in D such that is a dominating set for any in then the dominating set is secure. If for every there exists such that is a resolving set, then the resolving set is secure. These four cardinality values are the metric dimension of $G$, the connected metric dimension of , the secure metric dimension of , and the connected domination metric dimension of G, respectively. They correspond to the cardinality of the smallest resolving set of , the minimal connected resolving set, the minimal secure resolving set, and the minimal connected domination resolving set. In this paper, we introduce the secure connected dominant metric dimension of some graphs such as triangular snake graph, path graph, star tree and alternate quadrilateral snake. In particular, we derive the explicit formulas for the subdivision of triangular snake graph, alternate triangular snake graph, total graph of cycle graph and bistar tree.

Maximum energy bicyclic graphs containing two odd cycles with one common vertex

Maximum energy bicyclic graphs containing two odd cycles with one common vertex