Graph-theoretic Properties Research Articles

Fast Algorithms for Computing Fixed-Length Round Trips in Real-World Street Networks

This paper proposes and evaluates algorithms for calculating round trips of a prescribed length on directed street networks. This problem has several real-world applications, such as designing jogging routes and cycling tours. In this work, we focus specifically on methods that avoid the need to download, process, and store large map databases. At the same time, we aim for our methods to be fast, accurate, and capable of handling a wide range of prescribed distances, from just a few meters to many kilometres. To achieve this, our overall strategy involves using a small number of calls to a suitable online mapping service to collect relevant structural information for the problem at hand. All remaining computations are then performed locally on the client. Empirically, we demonstrate that our suggested techniques outperform existing open-source algorithms in terms of both accuracy and runtime requirements. Our most successful approach is based on multi-objective local search, utilizing specialized neighbourhood operators that exploit the underlying graph-theoretical properties of this problem, resulting in runtimes of around 2–3 s on a typical desktop computer.

Some results on the total zero-divisor graph of a commutative ring

PurposeThe purpose of this paper is to characterize a commutative ring R with identity which is not an integral domain such that ZT(R), the total zero-divisor graph of R is connected and to determine the diameter and radius of ZT(R) whenever ZT(R) is connected. Also, the purpose is to generalize some of the known results proved by Duric et al. on the total zero-divisor graph of R.Design/methodology/approachWe use the methods from commutative ring theory on primary decomposition and strong primary decomposition of ideals in commutative rings. The structure of ideals, the primary ideals, the prime ideals, the set of zero-divisors of the finite direct product of commutative rings is used in this paper. The notion of maximal Nagata prime of the zero-ideal of a commutative ring is also used in our discussion.FindingsFor a commutative ring R with identity, ZT(R) is the intersection of the zero-divisor graph of R and the total graph of R induced by the set of all non-zero zero-divisors of R. The zero-divisor graph of R and the total graph of R induced by the set of all non-zero zero-divisors of R are well studied. Hence, we determine necessary and sufficient condition so that ZT(R) agrees with the zero-divisor graph of R (respectively, agrees with the total graph induced by the set of non-zero zero-divisors of R). If Z(R) is an ideal of R, then it is noted that ZT(R) agrees with the zero-divisor graph of R. Hence, we focus on rings R such that Z(R) is not an ideal of R. We are able to characterize R such that ZT(R) is connected under the assumptions that the zero ideal of R admits a strong primary decomposition and Z(R) is not an ideal of R. With the above assumptions, we are able to determine the domination number of ZT(R).Research limitations/implicationsDuric et al. characterized Artinian rings R such that ZT(R) is connected. In this paper, we extend their result to rings R such that the zero ideal of R admits a strong primary decomposition and Z(R) is not an ideal of R. As an Artinian ring is isomorphic to the direct product of a finite number of Artinian local rings, we try to characterize R such that ZT(R) is connected under the assumption that R is ta finite direct product of rings R1, R2, … Rn with Z(Ri) is an ideal of Ri for each i between 1 to n. Their result on domination number of ZT(R) is also generalized in this paper. We provide several examples to illustrate our results proved.Practical implicationsThe implication of this paper is that the existing result of Duric et al. is applicable to large class of commutative rings thereby yielding more examples. Moreover, the results proved in this paper make us to understand the structure of commutative rings better. It also helps us to learn the interplay between the ring-theoretic properties and the graph-theoretic properties of the graph associated with it.Originality/valueThe results proved in this paper are original and they provide more insight into the structure of total zero-divisor graph of a commutative ring. This paper provides several examples. Not much work done in the area of total zero-divisor graph of a commutative ring. This paper is a contribution to the area of graphs and rings and may inspire other researchers to study the total zero-divisor graph in further detail.

Performance paradox of dynamic matching models under greedy policies

We consider the stochastic matching model on a non-bipartite compatibility graph and analyze the impact of adding an edge to the expected number of items in the system. One may see adding an edge as increasing the flexibility of the system, for example, asking a family registering for social housing to list fewer requirements in order to be compatible with more housing units. Therefore, it may be natural to think that adding edges to the compatibility graph will lead to a decrease in the expected number of items in the system and the waiting time to be assigned. In our previous work, we proved this is not always true for the First Come First Matched discipline and provided sufficient conditions for the existence of the performance paradox: despite a new edge in the compatibility graph, the expected total number of items can increase. These sufficient conditions are related to the heavy-traffic assumptions in queueing systems. The intuition behind this is that the performance paradox occurs when the added edge in the compatibility graph disrupts the draining of a bottleneck. In this paper, we generalize this performance paradox result to a family of so-called greedy matching policies and explore the type of compatibility graphs where such a paradox occurs. Intuitively, a greedy matching policy never leaves compatible items unassigned, so the state space of the system consists of finite words of item classes that belong to an independent set of the compatibility graph. Some examples of greedy matching policies are First Come First Match, Match the Longest, Match the Shortest, Random, Priority. We prove several results about the existence of performance paradoxes for greedy disciplines for some family of graphs. More precisely, we prove several results about the lifting of the paradox from one graph to another one. For a certain family of graphs, we prove that there exists a paradox for the whole family of greedy policies. Most of these results are based on strong aggregation of Markov chains and graph theoretical properties.

Rigidity on quantum symmetry for a certain class of graph C*-algebras

Quantum symmetry of graph C*-algebras has been studied, under the consideration of different formulations, in the past few years. It is already known that the compact quantum group (C(S1)∗C(S1)∗⋯∗C(S1)︸|E(Γ)|−times,Δ) [in short, *|E(Γ)|C(S1),Δ] always acts on a graph C*-algebra for a finite, connected, directed graph Γ in the category introduced by Joardar and Mandal, where |E(Γ)| ≔ number of edges in Γ. In this article, we show that for a certain class of graphs including Toeplitz algebra, quantum odd sphere, matrix algebra etc. the quantum symmetry of their associated graph C*-algebras remains *|E(Γ)|C(S1),Δ in the category as mentioned before. More precisely, if a finite, connected, directed graph Γ satisfies the following graph theoretic properties: (i) there does not exist any cycle of length ≥2 (ii) there exists a path of length (|V(Γ)| − 1) which consists all the vertices, where |V(Γ)| ≔ number of vertices in Γ (iii) given any two vertices (may not be distinct) there exists at most one edge joining them, then the universal object coincides with *|E(Γ)|C(S1),Δ. Furthermore, we have pointed out a few counter examples whenever the above assumptions are violated.

The Limiting Behaviors of the Gutman and Schultz Indices in Random 2k-Sided Chains

The study of complex networks with topological indices has flourished in recent years. The aim of this paper is to study the limiting behaviors of Gutman and Schultz indices in random polygonal chains, whose graph-theoretic mathematical properties and their future applications have attracted the interest of scientists. By applying the concepts of symmetry and asymptotics as well as the knowledge of probability theory, we obtain explicit analytic expressions for the Gutman and Schultz indices of n random 2k-vertex chains and prove that they converge to a normal distribution, which contributes to a deeper understanding of the structural features of random polygonal chains and plays a crucial role in the study of the limiting behavior of topological indices and their applications.

Characterizing Finite Groups through Equitable Graphs: A Graph-Theoretic Approach

This paper introduces equitable graphs of Type I associated with finite groups. We investigate the connectedness and some graph-theoretic properties of these graphs for various groups. Furthermore, we establish the novel concepts of the equitable square-free number and the equitable group. Our study includes an analysis of the equitable graphs for specific equitable groups. Additionally, we determine the first, second and forgotten Zagreb topological indices for the equitable graphs of Type I constructed from certain groups. Finally, we derive the adjacency matrix for this graph type built from cyclic p-groups.

On the order types of hammocks for domestic string algebras

In the representation-theoretic study of finite dimensional associative algebras over an algebraically closed field, Brenner introduced certain partially ordered sets known as hammocks to encode factorizations of maps between indecomposable finitely generated modules. In the context of domestic string algebras, Schröer introduced a simpler version of hammocks in his doctoral thesis that are bounded discrete linear orders. In this paper, we characterize the class of order types(=order isomorphism classes) of hammock linear orders for domestic string algebras as the bounded discrete ones amongst the class LOfp of finitely presented linear orders–the smallest class of linear orders containing finite linear orders as well as ω, and that is closed under isomorphisms, order reversal, finite order sums and antilexicographic products.In fact, we provide a multi-step algorithm to compute the order type of any closed interval in the hammock, and prove the correctness of this algorithm. A major step of this algorithm is the construction of a variation, which we call the arch bridge quiver, of a finite combinatorial gadget called the bridge quiver introduced by Schröer. He utilised the graph-theoretic properties of the bridge quiver for the computation of some representation-theoretic numerical invariants of domestic string algebras. The vertices of the bridge quiver are (representatives of cyclic permutations of) bands and its arrows are certain band-free strings. There is a natural but ill-behaved partial binary operation, ∘, on a superset of the set of bridges consisting of weak bridges such that bridges are precisely the ∘-irreducibles. We equip an even larger yet finite set of weak arch bridges with another partial binary operation, ∘H, to obtain a finite category. The binary operation ∘H uses isomorphisms between hammocks and explicitly relies on the description of the domestic string algebra as a bound quiver algebra. Each weak arch bridge admits a unique ∘H-factorization into arch bridges, i.e., the ∘H-irreducibles.

Characterization of rings with planar, toroidal or projective planar prime ideal sum graphs

Let R be a commutative ring with unity. The prime ideal sum graph PIS ( R ) of the ring R is the simple undirected graph whose vertex set is the set of all nonzero proper ideals of R and two distinct vertices I and J are adjacent if and only if I + J is a prime ideal of R. In this paper, we study some interplay between algebraic properties of rings and graph-theoretic properties of their prime ideal sum graphs. In this connection, we classify non-local commutative Artinian rings R such that PIS ( R ) is of crosscap at most two. We prove that there does not exist a non-local commutative Artinian ring whose prime ideal sum graph is projective planar. Further, we classify non-local commutative Artinian rings of genus one prime ideal sum graphs.

Connectome embedding in multidimensional graph spaces

Abstract Connectomes' topological organization can be quantified using graph theory. Here, we investigated brain networks in higher dimensional spaces defined by up to ten graph-theoretic nodal properties. These properties assign a score to nodes, reflecting their meaning in the network. Using 100 healthy unrelated subjects from the Human Connectome Project, we generated various connectomes (structural/functional, binary/weighted). We observed that nodal properties are correlated (i.e., they carry similar information) at whole-brain and subnetwork level. We conducted an exploratory machine learning analysis to test whether high dimensional network information differs between sensory and association areas. Brain regions of sensory and association networks were classified with an 80-86% accuracy in a 10D space. We observed the largest gain in machine learning accuracy going from a 2D to 3D space, with a plateauing accuracy towards 10D space, and nonlinear Gaussian kernels outperformed linear kernels. Finally, we quantified the Euclidean distance between nodes in a 10D graph space. The multidimensional Euclidean distance was highest across subjects in the default mode network (in structural networks) and frontoparietal and temporal lobe areas (in functional networks). To conclude, we propose a new framework for quantifying network features in high-dimensional spaces that may reveal new network properties of the brain.

Alterations of brain structural and functional connectivity networks topology and decoupling in pediatric myelin oligodendrocyte glycoprotein antibody-associated disease

ObjectiveTo investigate the alteration in structural and functional connectivity networks (SCN and FCN) as well as their coupling in pediatric myelin oligodendrocyte glycoprotein antibody-associated disease (MOGAD), and determine if these properties could serve as potential biomarkers for the disease. Materials and methodsIn total of 32 children with MOGAD and 30 age- and sex-matched healthy controls (HC) were employed to construct the SCN and FCN, respectively. The graph-theoretical analyses of the global properties, node properties of the 90 brain nodes, and the structural-functional connectivity (SC-FC) coupling of the two networks were performed. The graph-theoretical properties that exhibited significant differences were analyzed using partial correlation analysis in conjunction with the clinical scales, including the expanded disability status scale (EDSS), modified Rankin scale (mRS), and pediatric cerebral performance category (PCPC) of the MOGAD group. Subsequently, a machine learning model was developed to discriminate between MOGAD and the HC group, aiming to explore the potential of these properties as biomarkers. ResultsThe SCN of the MOGAD group exhibited aberrant global properties, including an increased characteristic path length (Lp) and a decreased global efficiency (Eg), along with reduced nodal properties such as degree centrality (Dc), nodal efficiency (Ne), and local efficiency in multiple nodes. The FCN of the MOGAD group only exhibited decreased Dc, Ne, and betweenness centrality in two nodes of nodal properties. Besides, MOGAD showed a significant decrease in SC-FC coupling compared to the HC group. The analysis of partial correlation revealed significant correlations between several properties and the scales of EDSS and mRS in the MOGAD group. The machine learning method was used to extract six features and establish the model, achieving a classification accuracy of 82.3% for MOGAD. ConclusionsPediatric MOGAD showed a more pronounced impairment in the SCN along with decoupling of SC-FC. Both partial correlation analysis and discriminant modeling suggest that alterations in brain network properties have the potential as biomarkers for assessing brain damage in MOGAD.

Topological alterations in white matter anatomical networks in cervical dystonia

BackgroundAccumulating neuroimaging evidence indicates that patients with cervical dystonia (CD) have changes in the cortico-subcortical white matter (WM) bundle. However, whether these patients’ WM structural networks undergo reorganization remains largely unclear. We aimed to investigate topological changes in large-scale WM structural networks in patients with CD compared to healthy controls (HCs), and explore the network changes associated with clinical manifestations.MethodsDiffusion tensor imaging (DTI) was conducted in 30 patients with CD and 30 HCs, and WM network construction was based on the BNA-246 atlas and deterministic tractography. Based on the graph theoretical analysis, global and local topological properties were calculated and compared between patients with CD and HCs. Then, the AAL-90 atlas was used for the reproducibility analyses. In addition, the relationship between abnormal topological properties and clinical characteristics was analyzed.ResultsCompared with HCs, patients with CD showed changes in network segregation and resilience, characterized by increased local efficiency and assortativity, respectively. In addition, a significant decrease of network strength was also found in patients with CD relative to HCs. Validation analyses using the AAL-90 atlas similarly showed increased assortativity and network strength in patients with CD. No significant correlations were found between altered network properties and clinical characteristics in patients with CD.ConclusionOur findings show that reorganization of the large-scale WM structural network exists in patients with CD. However, this reorganization is attributed to dystonia-specific abnormalities or hyperkinetic movements that need further identification.

Reconfigurations of Dynamic Functional Network Connectivity in Large-scale Brain Network after Prolonged Abstinence in Heroin Users.

Brain recovery phenomenon after long-term abstinence had been reported in substance use disorders. Yet, few longitudinal studies have been conducted to observe the abnormal dynamic functional connectivity (dFNC) of large-scale brain networks and recovery after prolonged abstinence in heroin users. The current study will explore the brain network dynamic connection reconfigurations after prolonged abstinence in heroin users (HUs). The 10-month longitudinal design was carried out for 40 HUs. The 40 healthy controls (HCs) were also enrolled. Group independent component analysis (GICA) and dFNC analysis were employed to detect the different dFNC patterns of addiction-related ICNs between HUs and HCs. The temporal properties and the graph-theoretical properties were calculated. Whether the abnormalities would be reconfigured in HUs after prolonged abstinence was then investigated. Based on eight functional networks extracted from GICA, four states were identified by the dFNC analysis. Lower mean dwell time and fraction rate in state4 were found for HUs, which were increased toward HCs after prolonged abstinence. In this state, HUs at baseline showed higher dFNC of RECN-aSN, aSN- aSN and dDMN-pSN, which decreased after protracted abstinence. A similar recovery phenomenon was found for the global efficiency and path length in abstinence HUs. Mean while, the abnormal dFNC strength was correlated with craving both at baseline and after abstinence. Our longitudinal study observed the large-scale brain network reconfiguration from the dynamic perspective in HUs after prolonged abstinence and improved the understanding of the neurobiology of prolonged abstinence in HUs.

Edge based metric dimension of various coffee compounds.

An important dietary source of physiologically active compounds, coffee also contains phenolic acids, diterpenes, and caffeine. According to a certain study, some coffee secondary metabolites may advantageously modify a number of anti-cancer defense systems. This research looked at a few coffee chemical structures in terms of edge locating numbers or edge metric size to better understand the mechanics of coffee molecules. Additionally, this research includes graph theoretical properties of coffee chemical structures. The chemicals found in coffee, such as caffeine, diterpene or cafestol, kahweol, chlorogenic, caffeic, gallotannins, and ellagitannins, are especially examined in these publications.

Resilient Time-Varying Formation Tracking for Heterogeneous High-Order Multiagent Systems With Multiple Dynamic Leaders

This paper investigates time-varying output formation tracking problems for heterogeneous multiagent systems in an adversarial environment, where both leaders and followers are vulnerable to cyber-attacks. Despite the presence of adversarial agents, the normal followers not suffering from cyber-attacks seek to achieve a desired for-mation configuration and track a safe trajectory generated by normal leaders' outputs simultaneously. To this end, a resilient time-varying output formation (RTVOF) tracking scheme is presented in this work. Firstly, to counteract the effect of adversarial agents, a safety strategy is proposed for general linear heterogeneous systems, which is a syn-thesis of the mean-subsequence-reduced family algorithms and an attack detection procedure. Secondly, using the neighboring interaction, a resilient distributed observer and a local formation controller are designed for each normal follower to estimate the state and track the output of the safe formation reference. Then, it is proved that the hetero-geneous multiagent system with multiple dynamic leaders and time-varying topologies can accomplish the RTVOF tracking if specific graph-theoretic properties and forma-tion feasibility conditions are satisfied. Finally, simulation examples are given to demonstrate the effectiveness of obtained results.

Crossing Numbers of the Cartesian Product of the Double Triangular Snake Graphs With Path Pm.

The crossing number Cr(G) of a graph G is the least number of edge crossings in all possible good drawings of G in the plane. Join and Cartesian products of graphs have many interesting graph-theoretical properties. In this paper, we evaluate the crossing number of the Cartesian product of double triangular snake graph DT2 with the path Pm. In this paper, we proved Cr(DT2 × Pm) = 6(m − 2), form ≥ 2 where Cr denotes the crossing number

Golden Laplacian Graphs

Many properties of the structure and dynamics of complex networks derive from the characteristics of the spectrum of the associated Laplacian matrix, specifically from the set of its eigenvalues. In this paper, we show that there exist graphs for which the ratio between the length of the spectrum (that is, the difference between the largest and smallest eigenvalues of the Laplacian matrix) and its spread (the difference between the second smallest eigenvalue and the smallest one) is equal to the golden ratio. We call such graphs Golden Laplacian Graphs (GLG). In this paper, we first find all such graphs with a number of nodes n≤10. We then prove several graph-theoretic and algebraic properties that characterize these graphs. These graphs prove to be extremely robust, as they have large vertex and edge connectivity along with a large isoperimetric constant. Finally, we study the synchronization properties of GLGs, showing that they are among the top synchronizable graphs of the same size. Therefore, GLGs represent very good candidates for engineering and communication networks.

Investigating robust associations between functional connectivity based on graph theory and general intelligence

Previous research investigating relations between general intelligence and graph-theoretical properties of the brain’s intrinsic functional network has yielded contradictory results. A promising approach to tackle such mixed findings is multi-center analysis. For this study, we analyzed data from four independent data sets (total N > 2000) to identify robust associations amongst samples between g factor scores and global as well as node-specific graph metrics. On the global level, g showed no significant associations with global efficiency or small-world propensity in any sample, but significant positive associations with global clustering coefficient in two samples. On the node-specific level, elastic-net regressions for nodal efficiency and local clustering yielded no brain areas that exhibited consistent associations amongst data sets. Using the areas identified via elastic-net regression in one sample to predict g in other samples was not successful for local clustering and only led to one significant, one-way prediction across data sets for nodal efficiency. Thus, using conventional graph theoretical measures based on resting-state imaging did not result in replicable associations between functional connectivity and general intelligence.

On definition of group homomorphism graph

In this paper, the group homomorphism graph is introduced and investigated. The group homomorphism graph, denoted by [Formula: see text], is an undirected graph in which the vertex set contains all homomorphisms excluding the monomorphisms and the zero homomorphism from the group [Formula: see text] to the group [Formula: see text], and two distinct vertices are adjacent if and only if the intersection of their kernels is non-trivial. We investigate the interplay between the graph-theoretic properties of this graph with algebraic properties of the groups. In this work, connectedness, diameter, clique number, chromatic number, domination number, independence number, etc. are found.

Graph theoretic properties of good sets in hypercube

A good set on k vertices is a vertex induced subgraph of the hypercube Q n that has the maximum number of edges. In this paper we discuss graph theoretic properties of good sets including connectivity, vertex decomposition, domination and identifying path covers of good sets.

Study on Leavitt path algebras and associated Grothendieck groups for directed enhanced power graphs

In this paper, we define directed enhanced power graphs for finite groups and study some graph-theoretic properties. Moreover, we investigate several algebraic properties of Leavitt path algebras of the directed enhanced power graphs and also of the directed punctured enhanced power graphs. We also compute Grothendieck groups of the Leavitt path algebras of directed punctured enhanced power graphs for cyclic groups and give partial answers to “the algebraic Kirchberg–Phillips question”.