Diameter Of Graph Research Articles

On the diameter of the zero-divisor graph over skew PBW extensions

The aim of this paper is to investigate the interplay between the algebraic properties of a skew Poincaré–Birkhoff–Witt extesion ring [Formula: see text] and the graph-theoretic properties of its zero-divisor graph. We are interested in studying the diameter of the zero-divisor graph of skew PBW extension rings. Among other results, we give a complete characterization of the possible diameters of [Formula: see text] in terms of the diameter of [Formula: see text].

Diameters of graphs on reduced words of $12$ and $21$-inflations

It is a classical result that any permutation in the symmetric group can be generated by a sequence of adjacent transpositions. The sequences of minimal length are called reduced words, and in this paper we study the graphs of these reduced words, with edges determined by relations in the underlying Coxeter group. Recently, the diameter has been calculated for the longest permutation $n\ldots 21$ by Reiner and Roichman as well as Assaf. In this paper we find inductive formulas for the diameter of the graphs of 12-inflations and many 21-inflations. These results extend to the associated graphs on commutation and long braid classes. Also, these results give a recursive formula for the diameter of the longest permutation, which matches that of Reiner, Roichman and Assaf. Lastly, We make progress on conjectured bounds of the diameter by Reiner and Roichman, which are based on the underlying hyperplane arrangement, and find families of permutations that achieve the upper bound and lower bound of the conjecture. In particular permutations that avoid 312 or 231 have graphs that achieve the upper bound.

Optimal strong approximation for quadrics over [formula omitted

Suppose q is a fixed odd prime power, F(x) is a non-degenerate quadratic form over Fq[t] of discriminant Δ in d≥5 variables x, and f,g∈Fq[t], λ∈Fq[t]d. We show that whenever deg⁡f≥(4+ε)deg⁡g+Oε,F(1), gcd⁡(Δ∞,fg)=O(1), and the necessary local conditions are satisfied, we have a solution x∈Fq[t]d to F(x)=f such that x≡λmodg. For d=4, we show that the same conclusion holds if we instead have deg⁡f≥(6+ε)deg⁡g+Oε,F(1). This gives us a new proof (independent of the Ramanujan conjecture over function fields proved by Drinfeld) that the diameter of any k-regular Morgenstern Ramanujan graph G is at most (2+ε)logk−1⁡|G|+Oε(1). In contrast to the d=4 case, our result is optimal for d≥5. Our main new contributions are a stationary phase theorem over function fields for bounding oscillatory integrals, and a notion of anisotropic cones to circumvent isotropic phenomena in the function field setting.

Spherical fullerene graphs that do not satisfy Andova and Škrekovski's conjecture

A fullerene graph is a planar, cubic, 3-connected graph with only pentagonal and hexagonal faces. In 2012, Andova and Škrekovski conjectured that the diameter of every fullerene graph with n vertices is at least √5n/3 - 1. They computed this lower bound by studying a particular class of fullerene graphs named spherical with icosahedral symmetry. We denote these graphs by Gi,j, by setting two parameters i,jε N*, such that i≤j. In their study, Andova and Škrekovski offered numerous properties of hexagonal lattices and calculated the diameter of two remarkable spherical fullerene graphs: G0,j and Gj,j. Although the conjecture is valid for these two distinct classes, it remains open deciding whether the premise is proper for all spherical fullerene graphs Gi,j.In this work, we present the first class of fullerene graphs with icosahedral symmetry that do not satisfy Andova and Skrekovski's conjecture, which refutes that this conjecture is valid for all spherical graphs. We also focus on showing properties of spherical fullerene graphs and the hexagonal lattice itself. We prove that all graphs Gi,j have a reduction of the form Gi-k,j-k, where k≤i, such that their triangular faces are entirely contained in the triangular faces of Gi,j. In addition, by setting k=i, this property states a particular link among Gi,j, Gi-1,j-1,• • •,G0,j-i, creating a chain of reductions of Gi,j, which implies that diam (Gi,j)≥diam (G0,j-i).

Some results on the open subset intersection graph of a product topological space

In the recent paper, R. A. Muneshwar et al., introduced a graph structure called open subset intersection graph g(t) on a topological space (X, t). In this paper, we study some important results of a graph g(t) of a product topological space (X × Y, t). We also determine relationship between diameter, girth, clique number, chromatic number, domination number etc. of an open subset intersection graph of a topological space (X × Y, t), (X, tX) and (Y, tY). Moreover, we proved that, if (X, tX) and (Y, tY) are discrete topological space then w(g(tX × tY)) = w(g(tX)) * w(g(tY)) – 2 and c(g(tX × tY)) = c(g(tX)) * c(g(tY)) – 2 and domination number of g(tX × tY) is 2. We also determine diameter and girth of intersection Graph of Product Topology on X × Y for different values of m and n.

On strongly r-ideals on commutative rings and some related graphs

A proper ideal I of a ring R is called an r-ideal if, whenever x, y ? R with xy ? I, we have x ? I or y ? Z(R) [R. Mohamadian, r-ideals in commutative rings, Turkish J. Math. 39(5) (2015),733-749]. In this article, we are interested in a subclass of the class of r-ideals which we call the class of strongly r-ideals. A proper ideal I of a ring R is called a strongly r-ideal if, whenever x, y ? R with xy ? I, we have x ? I or y ? Z(I). First, we give a basic study of this new concept which includes, among others, characterizations, properties and examples. After that, we use the introduced concept to characterize rings for which the diameter of the zero-divisor graph is less than or equal to two, rings for which the annihilator graph is complete, and rings for which the zero-annihilator graph is empty.

The missing Moore graph as an optimization problem

It has been an open question for 6 decades whether a Moore graph of diameter 2 and degree 57 exists. In this paper the question is posed as an optimization problem and an algorithm is described. The algorithm converges to solutions which are massively short of the number of edges required. This, and other supporting work, tend to suggest that the graph does not exist. The formulation presented is a particularly hard testbed for optimization algorithms. It is left as a challenge to others to develop alternative algorithms that may support the claim, or find solutions with more edges, or even construct the Moore graph.

Q-полиномиальный граф с массивом пересечений {60, 45, 8; 1, 12, 50} не существует

When studying amply regular graphs Γ of diameter d, in which for some vertex a the pair (Γd(a), Γd-1(a)) is a 2-scheme, it is proved that the subgraph induced by the set of points is a clique, coclique, or strongly regular graph. For a graph of diameter 3, it is established that this construction is a 2-scheme for any vertex a if and only if the graph is distance-regular and for any vertex a the subgraph Γ3(a) is a clique, coclique, or strongly regular graph (A.L. Gavrilyuk, A.A. Makhnev). An interesting question is whether there is a distance-regular graph with intersection array {60,45,8;1,12,50} such that Γ3(a) can be a 6×6-lattice and the pair (Γ3(a), Γ2(a)) will be a 2-scheme. In the paper of I.N. Belousov and A.A. Makhnev (2018) published a proof of the non-existence of the abovementioned graph, that contained errors. In this paper, we give a correct proof of this result.

The least Euclidean distortion constant of a distance-regular graph

In 2008, Vallentin made a conjecture involving the least distortion of an embedding of a distance-regular graph into Euclidean space. Vallentin’s conjecture implies that for a least distortion Euclidean embedding of a distance-regular graph of diameter d , the most contracted pairs of vertices are those at distance d . In this paper, we confirm Vallentin’s conjecture for several families of distance-regular graphs. We also provide counterexamples to this conjecture, where the largest contraction occurs between pairs of vertices at distance d − 1 . We suggest three alternative conjectures and prove them for various families of distance-regular graphs and for distance-regular graphs of diameter 3.

Synchronization of the generalized Kuramoto model with time delay and frustration

<abstract><p>We studied the collective behaviors of the time-delayed Kuramoto model with frustration under general network topology. For the generalized Kuramoto model with the graph diameter no greater than two and under a sufficient regime in terms of small time delay and frustration and large coupling strength, we showed that the complete frequency synchronization occurs exponentially fast when the initial configuration is distributed in a half circle. We also studied a complete network, which is a small perturbation of all-to-all coupling, as well as presented sufficient frameworks leading to the exponential emergence of frequency synchronization for the initial data confined in a half circle.</p></abstract>

On Computing the Diameter of (Weighted) Link Streams

A weighted link stream is a pair ( V , 𝔼) comprising V , the set of nodes, and 𝔼, the list of temporal edges ( u,v,t ,λ) , where u,v are two nodes in V , t is the starting time of the temporal edge, and λ is its travel time. By making use of this model, different notions of diameter can be defined, which refer to the following distances: earliest arrival time, latest departure time, fastest time, and shortest time. After proving that any of these diameters cannot be computed in time sub-quadratic with respect to the number of temporal edges, we propose different algorithms (inspired by the approach used for computing the diameter of graphs) that allow us to compute, in practice very efficiently, the diameter of quite large real-world weighted link stream for several definitions of the diameter. In the case of the fastest time distance and of the shortest time distance, we introduce the notion of pivot-diameter, to deal with the fact that temporal paths cannot be concatenated in general. The pivot-diameter is the diameter restricted to the set of pair of nodes connected by a path that passes through a pivot (that is, a node at a given time instant). We prove that the problem of finding an optimal set of pivots, in terms of the number of pairs connected, is NP-hard, and we propose and experimentally evaluate several simple and fast heuristics for computing “good” pivot sets. All the proposed algorithms (for computing either the diameter or the pivot-diameter) require very often a very low number of single source (or target) best path computations. We verify the effectiveness of our approaches by means of an extensive set of experiments on real-world link streams. We also experimentally prove that the temporal version of the well-known 2-sweep technique, for computing a lower bound on the diameter of a graph, is quite effective in the case of weighted link stream, by returning very often tight bounds.

Oriented diameter of graphs with given girth and maximum degree

Dankelmann, Guo and Surmacs proved that every bridgeless graph G of order n with given maximum degree Δ(G) has an orientation of diameter at most n−Δ(G)+3 [J. Graph Theory, 88(1)(2018), 5-17]. They also constructed a family of bridgeless graphs whose oriented diameter reaches this upper bound. In this paper, we show that G has an orientation of diameter at most n−⌊g(G)−12⌋(Δ(G)−4)−1, where g(G) is the girth of G. Moreover, we construct several families of bridgeless graphs whose oriented diameter attains n−⌊g(G)−12⌋(Δ(G)−4)−1, and prove that the upper bound is tight for Δ(G)≥4. We also give a necessary condition for a bridgeless graph to attain this upper bound. Furthermore, we verify that if G is a 3-connected graph with girth at least 5, then the oriented diameter of such G is at most n−⌊g(G)−12⌋(Δ(G)−4)−2.

Mean-field models of dynamics on networks via moment closure: An automated procedure.

In the study of dynamics on networks, moment closure is a commonly used method to obtain low-dimensional evolution equationsamenable to analysis. The variables in the evolution equationsare mean counts of subgraph states and are referred to as moments. Due to interaction between neighbors, each moment equationis a function of higher-order moments, such that an infinite hierarchy of equationsarises. Hence, the derivation requires truncation at a given order and an approximation of the highest-order moments in terms of lower-order ones, known as a closure formula. Recent systematic approximations have either restricted focus to closed moment equationsfor SIR epidemic spreading or to unclosed moment equationsfor arbitrary dynamics. In this paper, we develop a general procedure that automates both derivation and closure of arbitrary order moment equationsfor dynamics with nearest-neighbor interactions on undirected networks. Automation of the closure step was made possible by our generalized closure scheme, which systematically decomposes the largest subgraphs into their smaller components. We show that this decomposition is exact if these components form a tree, there is independence at distances beyond their graph diameter, and there is spatial homogeneity. Testing our method for SIS epidemic spreading on lattices and random networks confirms that biases are larger for networks with many short cycles in regimes with long-range dependence. A Mathematica package that automates the moment closure is available for download.

Plane graphs of diameter two are 2-optimal

Let G be a connected graph with n vertices. Suppose a fire breaks out at some vertex of G. At each time interval, firefighters can protect up to k vertices, and then the fire spreads to all unprotected neighbors. Let dnk(v) denote the minimum number of vertices that the fire may damage when a fire breaks out at vertex v. The k-expected damage of G, denoted by εk(G), is the expectation of the proportion of vertices that can be damaged from the fire, if the starting vertex of the fire is chosen uniformly at random, i.e., εk(G)=∑v∈V(G)dnk(v)/n2. A class of graphs G is called k-optimal if εk(G) tends to 0 as n tends to infinity for any G∈G. In this paper, we prove that planar graphs of diameter two are 2-optimal, which is the best possible.

Maximizing Convergence Time in Network Averaging Dynamics Subject to Edge Removal

We consider the consensus interdiction problem (CIP), in which the goal is to maximize the convergence time of consensus averaging dynamics subject to removing a limited number of network edges. We first show that CIP can be cast as an effective resistance interdiction problem (ERIP), in which the goal is to remove a limited number of network edges to maximize the effective resistance between a source node and a sink node. We show that ERIP is strongly NP-hard, even for bipartite graphs of diameter three with fixed source/sink edges, and establish the same hardness result for the CIP. We then show that both ERIP and CIP cannot be approximated up to a (nearly) polynomial factor assuming the exponential time hypothesis. Subsequently, we devise a polynomial-time $mn$-approximation algorithm for the ERIP that only depends on the number of nodes $n$ and the number of edges $m$ but is independent of the size of edge resistances. Finally, using a quadratic program formulation for the CIP, we devise an iterative approximation algorithm to find a first-order stationary solution for the CIP and evaluate its good performance through numerical experiments.

Linear Relationship between Mean‐Square Radius of Gyration and Graph Diameter, and Its Application to Network Polymers

AbstractMean‐square radius of gyration Rg2 and the graph diameter D of the random crosslinked network polymers are investigated to find a linear relationship, Rg2 = a D. The proportionality coefficient, a is dominated by the cycle (circuit) rank, or the number of intramolecular crosslinks kc, and a convenient equation is proposed for the relationship between a and kc. This relationship makes it possible to estimate Rg2 based on D and kc, which can reduce the required computational time to determine the Rg2‐values greatly. This new method is applied to find that the contraction factor g decreases with kc, and the differences in the primary chain length distribution that constitute the network polymers vanish for large kc‐values.

Diameter, Eccentricities and Distance Oracle Computations on H-Minor Free Graphs and Graphs of Bounded (Distance) Vapnik–Chervonenkis Dimension

Under the strong exponential-time hypothesis, the diameter of general unweighted graphs cannot be computed in truly subquadratic time (in the size of the input), as shown by Roditty and Williams. Nevertheless there are several graph classes for which this can be done such as bounded-treewidth graphs, interval graphs, and planar graphs, to name a few. We propose to study unweighted graphs of constant distance Vapnik–Chervonenkis (VC)-dimension as a broad generalization of many such classes—where the distance VC-dimension of a graph is defined as the VC-dimension of its ball hypergraph whose hyperedges are the balls of all possible radii and centers in . In particular for any fixed , the class of -minor free graphs has distance VC-dimension at most . Our first main result is a Monte Carlo algorithm that on graphs of distance VC-dimension at most , for any fixed , either computes the diameter or concludes that it is larger than in time , where only depends on and the notation suppresses polylogarithmic factors. We thus obtain a truly subquadratic-time parameterized algorithm for computing the diameter on such graphs. Then as a byproduct of our approach, we get a truly subquadratic-time randomized algorithm for constant diameter computation on all the nowhere dense graph classes. The latter classes include all proper minor-closed graph classes, bounded-degree graphs, and graphs of bounded expansion. Before our work, the only known such algorithm was resulting from an application of Courcelle’s theorem; see Grohe, Kreutzer, and Siebertz [J. ACM, 64 (2017), pp. 1–32]. For any graph of constant distance VC-dimension, we further prove the existence of an exact distance oracle in truly subquadratic space, that answers distance queries in truly sublinear time (in the number of vertices). The latter generalizes prior results on proper minor-closed graph classes to a much larger graph class. Finally, we show how to remove the dependency on for any graph class that excludes a fixed graph as a minor. More generally, our techniques apply to any graph with constant distance VC-dimension and polynomial expansion (or equivalently having strongly sublinear balanced separators). As a result for all such graphs one obtains a truly subquadratic-time deterministic algorithm for computing all the eccentricities, and thus both the diameter and the radius. Our approach can be generalized to the -minor free graphs with bounded positive integer weights. We note that all our algorithms for the diameter problem can be adapted for computing the radius, and more generally all the eccentricities. Our approach is based on the work of Chazelle and Welzl who proved the existence of spanning paths with strongly sublinear stabbing number for every hypergraph of constant VC-dimension. We show how to compute such paths efficiently by combining known algorithms for the stabbing number problem with a clever use of -nets, region decomposition, and other partition techniques.

The configuration space of a robotic arm over a graph

In this paper, we investigate the configuration space [Formula: see text] associated with the movement of a robotic arm of length [Formula: see text] on a grid over an underlying graph [Formula: see text], anchored at a vertex [Formula: see text]. We study an associated poset with inconsistent pairs (PIP) [Formula: see text] consisting of indexed paths on [Formula: see text]. This PIP acts as a combinatorial model for the robotic arm, and we use [Formula: see text] to show that the space [Formula: see text] is a CAT(0) cubical complex, generalizing work of Ardila, Bastidas, Ceballos, and Guo. This establishes that geodesics exist within the configuration space, and yields explicit algorithms for moving the robotic arm between different configurations in an optimal fashion. We also give a tight bound on the diameter of the robotic arm transition graph — the maximal number of moves necessary to change from one configuration to another — and compute this diameter for a large family of underlying graphs [Formula: see text].

A spatially implicit model fails to predict the structure of spatially explicit metacommunities under high dispersal

Metacommunities are the product of species dispersal and topology. Metacommunity studies often use spatially implicit models, implemented by fully connected topologies, in which the precise spatial arrangement of habitat patches is not specified. Few studies use spatially explicit models, even though real-world metacommunities are likely structured by topology. Here, we test whether a spatially implicit resource consumption model based on a fully connected topology could predict the structure of spatially explicit metacommunities. Having controlled for environmental heterogeneity, we focus specifically on the effects of species dispersal and topology on metacommunity structure. We classified the topologies according to the shortest path between the most distant nodes (i.e. the graph diameter). Topologies with small diameters are tightly connected, whereas large diameter graphs are loosely connected. Some general trends emerged with increasing dispersal rate, such as a hump-shaped pattern in α-diversity, and a plateau followed by a decline in γ- diversity. However, the importance of topology was also apparent: α-diversity peaked at low dispersal rates in small diameter topologies, but at high dispersal rates in large diameter topologies. At low dispersal rates, α-diversity was higher in spatially implicit than in spatially explicit metacommunities. At medium dispersal we detected stronger species sorting in the small diameter than in the large diameter topologies. Increasing dispersal caused α-diversity to decline more dramatically in small diameter topologies. Smaller metacommunities were dominated by regional competitors, whereas larger communities exhibited patterns of species biomass distribution leading to emergent niche structures. Increasing dispersal caused the mean productivity of each patch to undergo partial declines in spatially implicit metacommunities but continue to decline sharply in spatially explicit metacommunities. We conclude that spatially implicit models should be used cautiously when predicting the biodiversity, community composition or ecosystem functions of spatially explicit metacommunities at medium, and especially at high dispersal rates.

Monitoring the edges of a graph using distances

We introduce a new graph-theoretic concept in the area of network monitoring. A set $M$ of vertices of a graph $G$ is a \emph{distance-edge-monitoring set} if for every edge $e$ of $G$, there is a vertex $x$ of $M$ and a vertex $y$ of $G$ such that $e$ belongs to all shortest paths between $x$ and $y$. We denote by $dem(G)$ the smallest size of such a set in $G$. The vertices of $M$ represent distance probes in a network modeled by $G$; when the edge $e$ fails, the distance from $x$ to $y$ increases, and thus we are able to detect the failure. It turns out that not only we can detect it, but we can even correctly locate the failing edge. In this paper, we initiate the study of this new concept. We show that for a nontrivial connected graph $G$ of order $n$, $1\leq dem(G)\leq n-1$ with $dem(G)=1$ if and only if $G$ is a tree, and $dem(G)=n-1$ if and only if it is a complete graph. We compute the exact value of $dem$ for grids, hypercubes, and complete bipartite graphs. Then, we relate $dem$ to other standard graph parameters. We show that $demG)$ is lower-bounded by the arboricity of the graph, and upper-bounded by its vertex cover number. It is also upper-bounded by twice its feedback edge set number. Moreover, we characterize connected graphs $G$ with $dem(G)=2$. Then, we show that determining $dem(G)$ for an input graph $G$ is an NP-complete problem, even for apex graphs. There exists a polynomial-time logarithmic-factor approximation algorithm, however it is NP-hard to compute an asymptotically better approximation, even for bipartite graphs of small diameter and for bipartite subcubic graphs. For such instances, the problem is also unlikey to be fixed parameter tractable when parameterized by the solution size.