In this paper, we mainly consider the Hamiltonian indices of three classes of graphs obtained from Petersen graph, that is, the minimum integer m of m-time iterated line graph Lm(G) of these three classes of graphs such that Lm(G) is Hamiltonian. We...
Given a graph G = (V, E), the edge set can be partitioned into k dimensions, for a positive integer k. The set of all i-dimensional edges of G is a subset of E(G) denoted by Ei. A Hamiltonian cycle C on G contains all vertices on G. Let Ei(C) = E(C)...
Recently, the problem of counting Hamiltonian cycles in 2-tiled graphs was resolved by Vegi Kalamar, Bokal, and Žerak. In this paper, we continue our research on generalized tiled graphs. We extend algorithms on counting traversing Hamiltonian c...
Let G=(V,E) be a graph. The first Zagreb index of a graph G is defined as ∑u∈VdG2(u), where dG(u) is the degree of vertex u in G. A graph G is called Hamiltonian (resp. traceable) if G has a cycle (resp. path) containing all the vertices of...
In this study, we investigate Hamiltonian cycles in the right-Cayley graphs of gyrogroups. More specifically, we give a gyrogroup version of the factor group lemma and show that some right-Cayley graphs of certain gyrogroups are Hamiltonian.
In 1930, Kuratowski showed that K3,3 and K5 are the only two minor-minimal nonplanar graphs. Robertson and Seymour extended finiteness of the set of forbidden minors for any surface. Širáň and Kochol showed that there are infinitely many k-crossing-c...
A graph is called Hamiltonian connected if it contains a Hamiltonian path between any two distinct vertices. In the past, we proved the Hamiltonian path and cycle problems for general supergrid graphs to be NP-complete. However, they are still open f...
The first Zagreb index of a graph G is defined as the sum of the squares of the degrees of all the vertices in G. The Laplacian spectral radius of a graph G is defined as the largest eigenvalue of the Laplacian matrix of the graph G. In this paper, w...
Finding a Hamiltonian cycle in a graph G = (V, E) is a well-known problem. The challenge of finding a Hamiltonian cycle that avoids these faults when faulty vertices or edges are present has been extensively studied. When the edge set of G is partiti...
Hamiltonian cycle problems play a central role in graph theory and have wide-ranging applications in network-on-chip architectures, interconnection networks, and large-scale parallel systems. When a network contains faulty nodes or faulty links, the...
The Hamiltonian cycle reconfiguration problem asks, given two Hamiltonian cycles C 0 and C t of a graph G, whether there is a sequence of Hamiltonian cycles C 0 , C 1 , … , C t such that C i can be obtaine...
The forgotten topological index of a (molecule) graph is the sum of cubes of all its vertex degrees, which plays a significant role in measuring the branching of the carbon atom skeleton. It is meaningful and difficult to explore sufficient condition...
The method of the optimal movement trajectory construction in the terrain patrolling tasks is proposed. The method is based on the search of the Hamiltonian circuit on the graph of the terrain map and allows automatic construction of the optimal clos...
Let R be a finite ring. In this paper, we mainly explore the conditions to ensure the graph BΓn defined by a system of equations {fi|i=2,…,n} to be a Cayley graph or a Hamiltonian graph. More precisely, we prove that BΓn is a Cayle...
We consider a compact metric graph of size ε and attach to it several edges (leads) of length of order one (or of infinite length). As ε goes to zero, the graph G ε obtained in this way looks like the star-...
Fuzzy Topological Topographic Mapping (FTTM) is a mathematical model that consists of a set of homeomorphic topological spaces designed to solve the neuro magnetic inverse problem. A sequence of FTTM, denoted as FTTMn, is an extension of FTTM that is...
We analyzed a special class of graph traversal problems, where the distances are stochastic, and the agent is restricted to take a limited range in one go. We showed that both constrained shortest Hamiltonian pathfinding problems and disassembly line...
Description of interacting spin systems relies on understanding the spectral properties of the corresponding spin Hamiltonians. However, the eigenvalue problems arising here lead to algebraic problems too complex to be analytically tractable. This is...
Aperiodic tessellations of polykite unitiles, such as hats and turtles, and the recently introduced hares, red squirrels, and gray squirrels, have attracted significant interest due to their structural and combinatorial properties. Our primary object...
In this paper, we explore the connection between sensor networks and graph theory. Sensor networks represent distributed systems of interconnected devices that collect and transmit data, while graph theory provides a robust framework for modeling and...
A graph is said to be an intersection graph if there is a set of objects such that each vertex corresponds to an object and two vertices are adjacent if and only if the corresponding objects have a nonempty intersection. There are several natural gra...
The longest (s,t)-path problem on supergrid graphs is known to be NP-complete. However, the complexity of this problem on supergrid graphs with or without holes is still unknown.In the past, we presented linear-time algorithms for solving the longest...
Let G be a simple graph of order n with vertex set V(G) and edge set E(G), and let k be an integer such that 1≤k≤n−1. The k-token graph G{k} of G is the graph whose vertices are the k-subsets of V(G), where two vertices A and B are adjacent in G{k} w...
Short or moderate-length non-binary low-density parity-check (NB-LDPC) codes have the potential applications in future low latency and high-reliability communication thanks to the strong error correction capability and parallel decoding. Because of t...
In the design of real-world networks, researchers evaluate various structural parameters to assess vulnerability, including connectivity, toughness, and tenacity. Recently, the tightness metric has emerged as a potentially superior vulnerability meas...
In loop quantum gravity (LQG), states of the gravitational field are represented by labeled graphs called spin networks. Their dynamics can be described by a Hamiltonian constraint, which acts on the spin network states, modifying both spins and grap...
This paper studies the Hamiltonian cycle problem (HCP) and the traveling salesman problem (TSP) on D-Wave quantum systems. Motivated by the fact that most libraries present their benchmark instances in terms of adjacency matrices, we develop a novel...
We consider the problem of measuring the similarity between two graphs using continuous-time quantum walks and comparing their time-evolution by means of the quantum Jensen-Shannon divergence. Contrary to previous works that focused solely on undirec...
This paper considers commuting graphs over the semidihedral group SD8n. We compute their eigenvalues and obtain that these commuting graphs are not hyperenergetic for odd n≥15 or even n≥2. We further compute the Laplacian spectrum, the Laplacia...
This research is focused on evolutionary algorithms, with genetic and memetic algorithms discussed in more detail. A graph theory problem related to finding a minimal Hamiltonian cycle in a complete undirected graph (Travelling Salesman Problem&mdash...
The balanced hypercube network, which is a novel interconnection network for parallel computation and data processing, is a newly-invented variant of the hypercube. The particular feature of the balanced hypercube is that each processor has its own b...
In graph theory, Hamiltonian path refers to the path that visits each vertex exactly once. In this paper, we designed a method to generate random Hamiltonian path within digital images, which is equivalent to permutation in image encryption. By these...
In this paper, we introduce and investigate an ideal-based dot total graph of commutative ring R with nonzero unity. We show that this graph is connected and has a small diameter of at most two. Furthermore, its vertex set is divided into three disjo...
A binary modified de Bruijn sequence is an infinite and periodic binary sequence derived by removing a zero from the longest run of zeros in a binary de Bruijn sequence. The minimal polynomial of the modified sequence is its unique least-degree chara...
A side skirt is a planar rooted tree T, T≠P2, where the root of T is a vertex of degree at least two, and all other vertices except the leaves are of degree at least three. A reduced Halin graph or a skirted graph is a plane graph G=T∪P, where...
To support doubly fed wind turbine (DFWT) groups in offshore wind farms, this paper proposes a distributed coordinated control based on the Hamiltonian energy theory. This strategy provides global stability to closed-loop systems and facilitates outp...
In 2014, some scholars showed that every 2-connected claw-free graph G with independence number α(G)≤3 is Hamiltonian with one exception of family of graphs. If a nontrivial path contains only internal vertices of degree two and end vertices...
Cayley and nilpotent graphs arise from the interaction between graph theory and algebra and are used to visualize the structures of some algebraic objects as groups and commutative rings. On the other hand, Green and Schroll introduced Brauer graph a...
Quantum search algorithms provide a way to speed up combinatorial search, and have found several applications in modern quantum technology. In particular, spatial search on graphs, based on continuous-time quantum walks (CTQW), represents a promising...
Trotter–Suzuki decompositions are frequently used in the quantum simulation of quantum chemistry. They transform the evolution operator into a form implementable on a quantum device, while incurring an error—the Trotter error. The Trotter...
Suppose that G is a graph over n vertices. G has n eigenvalues (of adjacency matrix) represented by λ1,λ2,⋯,λn. The Gaussian Estrada index, denoted by H(G) (Estrada et al., Chaos 27(2017) 023109), can be defined as H(G)=∑i=1n...
The interconnection network represents an interconnected structure of processors that strongly determines the performance quality of a parallel processing system. The shuffle-exchange permutation (SEP) network with three degrees has high fault tolera...
In practical applications, the temporal completeness of knowledge graphs is of great importance. However, previous studies have mostly focused on static knowledge graphs, generally neglecting the dynamic evolutionary properties of facts. Moreover, th...
Embedding cycles into a network topology is crucial for a network simulation. In particular, embedding Hamiltonian cycles is a major requirement for designing good interconnection networks. A graph G is called r-spanning cyclable if, for any r distin...
Supergrid graphs are derived by computing stitch paths for computerized embroidery machines. In the past, we have studied the Hamiltonian-related properties of supergrid graphs and their subclasses of graphs. In this paper, we propose a generalized g...
We propose a dynamic extension of the Petford–Welsh coloring algorithm that estimates the chromatic number of a graph without requiring k as an input. The basic algorithm is based on the model that is closely related to the Boltzmann machines t...
One of the important issues in evaluating an interconnection network is to study the hamiltonian cycle embedding problems. A graph G is spanning k-edge-cyclable if for any k independent edges e1,e2,…,ek of G, there exist k vertex-disjoint cycl...
A graph with n vertices is called an n-graph. A spanning tree with at most k leaves is referred to as a spanning k-ended tree. Spanning k-ended trees are important in various fields such as network design, graph theory, and communication networks. Th...
The Sustainable Time-Dependent Cheapest Path Problem (STDCPP) entails locating a Hamiltonian path that covers all of the graph’s vertices at the lowest possible total sustainability cost. The issue is inspired by actual city logistics, where it...
The existence of chaos and the quest of dense orbits have been recently considered for dynamical systems given by multivalued linear operators. We consider the notions of topological transitivity, topologically mixing property, hypercyclicity, period...
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