Essentially 3-edge-connected reduced graph of diameter three

In a graph, a matching cut is an edge cut that is a matching. Matching Cut is the problem of deciding whether or not a given graph has a matching cut, which is known to be NP-complete even when restricted to bipartite graphs. It has been proved that Matching Cut is polynomially solvable for graphs of diameter two. In this paper, we show that, for any fixed integer d geq 4, Matching Cut is NP-complete in the class of graphs of diameter d. This almost resolves an open problem posed by Borowiecki and Jesse-Jozefczyk in [Matching cutsets in graphs of diameter 2, Theoretical Computer Science 407 (2008) 574-582]. We then show that, for any fixed integer d geq 5, Matching Cut is NP-complete even when restricted to the class of bipartite graphs of diameter d. Complementing the hardness results, we show that Matching Cut is in polynomial-time solvable in the class of bipartite graphs of diameter at most three, and point out a new and simple polynomial-time algorithm solving Matching Cut in graphs of diameter 2.

We provide a process to extend any bipartite diametrical graph of diameter 4 to an -graph of the same diameter and partite sets. For a bipartite diametrical graph of diameter 4 and partite sets and , where , we prove that is a sharp upper bound of and construct an -graph in which this upper bound is attained, this graph can be viewed as a generalization of the Rhombic Dodecahedron. Then we show that for any , the graph is the unique (up to isomorphism) bipartite diametrical graph of diameter 4 and partite sets of cardinalities and , and hence in particular, for , the graph which is just the Rhombic Dodecahedron is the unique (up to isomorphism) bipartite diametrical graph of such a diameter and cardinalities of partite sets. Thus we complete a characterization of -graphs of diameter 4 and cardinality of the smaller partite set not exceeding 6. We prove that the neighborhoods of vertices of the larger partite set of form a matroid whose basis graph is the hypercube . We prove that any -graph of diameter 4 is bipartite self complementary, thus in particular . Finally, we study some additional properties of concerning the order of its automorphism group, girth, domination number, and when being Eulerian.

For a connected graph [Formula: see text], we use the notation [Formula: see text] to represent the distance between two node points [Formula: see text] and [Formula: see text] and it is the minimum of the lengths of all paths between them. The eccentricity [Formula: see text] of a node point [Formula: see text] is considered as the maximum length of all shortest paths starts from [Formula: see text] to the remaining nodes, i.e., [Formula: see text]. The diameter of a graph [Formula: see text], we denote it by [Formula: see text] and it is the length of the longest shortest path in [Formula: see text], i.e., [Formula: see text]. Also, the radius of a graph [Formula: see text], we denote it by the symbol [Formula: see text] and it is the least eccentricity of all node points in [Formula: see text], i.e., [Formula: see text]. The central vertex/node point [Formula: see text] of a graph [Formula: see text] is a node whose eccentricity is same as [Formula: see text]’s radius, i.e., [Formula: see text]. The collection of all central nodes of a graph [Formula: see text] is considered as the center of [Formula: see text] and it is symbolized by [Formula: see text], i.e., [Formula: see text]. A graph may have one or more central vertices. This paper develops an optimal algorithm to compute the diameter, radius and central node (s) of the permutation graph having [Formula: see text] node points in [Formula: see text] time. We have also established a tight relation between radius and diameter of permutation graphs.

Let G = (V, E) be a graph. The distance d (u, v) between two vertices u and v is the length of the shortest path between them. The diameter of the graph is the length of the longest path of the shortest paths between any two graph vertices (u ,v) of a graph, . In this paper we propose algorithms for finding diameter of a hierarchy graph using DFS. Diameter of the hierarchy graph using DFS algoritm is four.

Vulnerability in graphs of diameter four

A distance‐regular graph of diameter d has 2d intersection numbers that determine many properties of graph (e.g., its spectrum). We show that the first six coefficients of the matching polynomial of a distance‐regular graph can also be determined from its intersection array, and that this is the maximum number of coefficients so determined. Also, the converse is true for distance‐regular graphs of small diameter—that is, the intersection array of a distance‐regular graph of diameter 3 or less can be determined from the matching polynomial of the graph.

In the pursuit of obtaining largest graphs of given maximum degree Δ and diameter D, many construction techniques have been developed. Compounding of graphs is one such technique. In this article, by means of the compounding of complete graphs into a bipartite Moore graph of diameter 6, we obtain a family of large graphs of the same diameter. For maximum degrees Δ = 5, 6, 9, 12, and 14, members of this family constitute the largest known graphs of diameter 6. © 2008 Wiley Periodicals, Inc. NETWORKS, 2009

One can model a social network as a long-range percolation model on a graph {0, 1, …, N}2. The edges (x, y) of this graph are selected with probability ≈ β/||x - ys if ||x - y|| > 1, and with probability 1 if ||x - y|| = 1, for some parameters β, s > 0. That is, people are more likely to be acquainted with their neighbors than with people at large distance. This model was introduced by Benjamini and Berger [2] and it resembles a model considered by Kleinberg in [6], [7]. We are interested in how small (probabilistically) is the diameter of this graph as a function of β and s, thus relating to the famous Milgram's experiment which led to the six degrees of separation concept. Extending the work by Benjamini and Berger, we consider a d-dimensional version of this question on a node set {0, 1, …, N}d and obtain upper and lower bounds on the expected diameter of this graph. Specifically, we show that the expected diameter experiences phase transitions at values s = d and s = 2d. We compare the algorithmic implication of our work to the ones of Kleinberg, [6].

In this paper, we study the configuration model (CM) with independent and identically-distributed (i.i.d.) degrees. We establish a phase transition for the diameter when the power-law exponent τ of the degrees satisfies τ ∈ (2, 3). Indeed, we show that for τ > 2 and when vertices with degree 1 or 2 are present with positive probability, the diameter of the random graph is, with high probability, bounded from below by a constant times the logarithm of the size of the graph. On the other hand, assuming that all degrees are 3 or more, we show that, for τ ∈ (2, 3), the diameter of the graph is, with high probability, bounded from above by a constant times the log log of the size of the graph.

Let Gbe a finite p-solvable group. Let us consider the graph Γ* p (G) whose vertices are the primes which occur as the divisors of the conjugacy classes of p-regular elements of G and two primes are joined by an edge if there exists such a class whose size is divisible by both primes. Suppose that Γ p *(G) is a connected graph, then we prove that the diameter of this graph is at most 3 and this is the best bound.

The diameter of a graph is the maximum distance among all pairs of nodes. Determining the diameter of a graph in the tradition way costs O(mn) time, where n is the number of nodes and m is the number of edges. A social network can be modelled as a graph. With the rapid expansion of social networks, the number of nodes in a social network could be hundreds of millions. In this paper, we propose a new approach for computing the diameters of large undirected unweighted graphs. The worst case time complexity is still O(mn). In practice, especially for social network graphs, the running time is O(m). Our approach is based on BFS to select a proper node as the starting node of a BFS process is the most important issue when computing the diameter. We show how to choose the good nodes with small cost.

We study the diameter of LPS Ramanujan graphs Xp,q. We show that the diameter of the bipartite Ramanujan graphs is greater than (4/3)logp(n)+O(1), where n is the number of vertices of Xp,q. We also construct an infinite family of (p+1)-regular LPS Ramanujan graphs Xp,m such that the diameter of these graphs is greater than or equal to ⌊(4/3)logp(n)⌋. On the other hand, for any k-regular Ramanujan graph we show that only a tiny fraction of all pairs of vertices have distance greater than (1+ϵ) logk–1(n). We also have some numerical experiments for LPS Ramanujan graphs and random Cayley graphs which suggest that the diameters are asymptotically (4/3)logk–1(n) and logk–1(n), respectively.

In graph theory and study of fault tolerance and transmission delay of networks, connectivity and diameter of a graph are two very important parameters and have been deeply studied by many authors. Wide diameter combining connectivity with diameter is a more important parameter to measure fault tolerance and efficiency of parallel processing computer networks and has received much attention in the recent years. Diameter with width k of a graph G is defined as the minimum integer d for which between any two distinct vertices in G there exist at least k internally disjoint paths of length at most d. In the present paper, the tight upper bounds of wide diameter of the Cartesian product graphs are obtained. Some known results can be deduced or improved from ours.

Small diameter neighbourhood graphs for the traveling salesman problem: at most four moves from tour to tour

The diameter of a graph is among its most basic parameters. Since a few years ago, it moreover became a key issue to compute it for massive graphs in the context of complex network analysis. However, known algorithms, including the ones producing approximate values, have too high a time and/or space complexity to be used in such cases. We propose here a new approach relying on very simple and fast algorithms that compute (upper and lower) bounds for the diameter. We show empirically that, on various real-world cases representative of complex networks studied in the literature, the obtained bounds are very tight (and even equal in some cases). This leads to rigorous and very accurate estimations of the actual diameter in cases which were previously untractable in practice.