Description:
In 1960, Hoffman and Singleton investigated the existence of Moore graphs of diameter 2 (graphs of maximum degree d and d² + 1 vertices), and found that such graphs exist only for d = 2; 3; 7 and possibly 57. In 1980, Erdös et al., using eigenvalue analysis, showed that, with the exception of C4, there are no graphs of diameter 2, maximum degree d and d² vertices. In this paper, we show that graphs of diameter 2, maximum degree d and d² - 1 vertices do not exist for most values of d with d ≥ 6, and conjecture that they do not exist for any d ≥ 6.
Description:
We consider bipartite graphs of degree A<2, diameter D = 3, and defect 2 (having 2 vertices less than the bipartite Moore bound). Such graphs are called bipartite (â–³,3, -2) -graphs. We prove the uniqueness of the known bipartite (3, 3, -2) -graph and bipartite (4, 3, -2)-graph. We also prove several necessary conditions for the existence of bipartite (â–³,3, -2) - graphs. The most general of these conditions is that either â–³ or â–³-2 must be a perfect square. Furthermore, in some cases for which the condition holds, in particular, when â–³ = 6 and â–³ = 9, we prove the non-existence of the corresponding bipartite (â–³,3,-2)-graphs, thus establishing that there are no bipartite (â–³,3, -2)-graphs, for 5
Description:
We consider bipartite graphs of degree A<2, diameter D = 3, and defect 2 (having 2 vertices less than the bipartite Moore bound). Such graphs are called bipartite (â–³,3, -2) -graphs. We prove the uniqueness of the known bipartite (3, 3, -2) -graph and bipartite (4, 3, -2)-graph. We also prove several necessary conditions for the existence of bipartite (â–³,3, -2) - graphs. The most general of these conditions is that either â–³ or â–³-2 must be a perfect square. Furthermore, in some cases for which the condition holds, in particular, when â–³ = 6 and â–³ = 9, we prove the non-existence of the corresponding bipartite (â–³,3,-2)-graphs, thus establishing that there are no bipartite (â–³,3, -2)-graphs, for 5
Description:
The Moore bound constitutes both an upper bound on the order of a graph of maximum degree d and diameter D = k and a lower bound on the order of a graph of minimum degree d and odd girth g = 2k + 1. Graphs missing or exceeding the Moore bound by ε are called graphs with defect or excess ε, respectively. While Moore graphs (graphs with ε = 0) and graphs with defect or excess 1 have been characterized almost completely, graphs with defect or excess 2 represent a wide unexplored area. Graphs with defect (excess) 2 satisfy the equation Gd,k(A) = Jn +B (Gd,k(A) = Jn - B), where A denotes the adjacency matrix of the graph in question, n its order, Jn the n × n matrix whose entries are all 1's, B the adjacency matrix of a union of vertex-disjoint cycles, and Gd,k(x) a polynomial with integer coefficients such that the matrix Gd,k(A) gives the number of paths of length at most k joining each pair of vertices in the graph. In particular, if B is the adjacency matrix of a cycle of order n we call the corresponding graphs graphs with cyclic defect or excess; these graphs are the subject of our attention in this paper. We prove the non-existence of infinitely many such graphs. As the highlight of the paper we provide the asymptotic upper bound of O(64/3 d3/2) for the number of graphs of odd degree d ≥ 3 and cyclic defect or excess. This bound is in fact quite generous, and as a way of illustration, we show the non-existence of some families of graphs of odd degree d ≥ 3 and cyclic defect or excess. Actually, we conjecture that, apart from the Möbius ladder on 8 vertices, no non-trivial graph of any degree ≥ 3 and cyclic defect or excess exists.
Description:
The Moore bound constitutes both an upper bound on the order of a graph of maximum degree d and diameter D = k and a lower bound on the order of a graph of minimum degree d and odd girth g = 2k + 1. Graphs missing or exceeding the Moore bound by ε are called graphs with defect or excess ε, respectively. While Moore graphs (graphs with ε = 0) and graphs with defect or excess 1 have been characterized almost completely, graphs with defect or excess 2 represent a wide unexplored area. Graphs with defect (excess) 2 satisfy the equation Gd,k(A) = Jn +B (Gd,k(A) = Jn - B), where A denotes the adjacency matrix of the graph in question, n its order, Jn the n × n matrix whose entries are all 1's, B the adjacency matrix of a union of vertex-disjoint cycles, and Gd,k(x) a polynomial with integer coefficients such that the matrix Gd,k(A) gives the number of paths of length at most k joining each pair of vertices in the graph. In particular, if B is the adjacency matrix of a cycle of order n we call the corresponding graphs graphs with cyclic defect or excess; these graphs are the subject of our attention in this paper. We prove the non-existence of infinitely many such graphs. As the highlight of the paper we provide the asymptotic upper bound of O(64/3 d3/2) for the number of graphs of odd degree d ≥ 3 and cyclic defect or excess. This bound is in fact quite generous, and as a way of illustration, we show the non-existence of some families of graphs of odd degree d ≥ 3 and cyclic defect or excess. Actually, we conjecture that, apart from the Möbius ladder on 8 vertices, no non-trivial graph of any degree ≥ 3 and cyclic defect or excess exists.
Description:
It is known that the Moore bipartite bound provides an upper bound on the order of a connected bipartite graph. In this paper we deal with bipartite graphs of maximum degree
Description:
It is known that the Moore bipartite bound provides an upper bound on the order of a connected bipartite graph. In this paper we deal with bipartite graphs of maximum degree
Description:
Using eigenvalue analysis, it was shown by Erdos et al. that, with the exception of C4, there are no graphs of diameter 2, maximum degree d and d2 vertices. In this paper, we show that graphs of diameter 2, maximum degree d and d2-1 vertices do not exist for most values of d, when d is even, and we conjecture that they do not exist for any even d greater than 4.
Description:
Using eigenvalue analysis, it was shown by Erdos et al. that, with the exception of C4, there are no graphs of diameter 2, maximum degree d and d2 vertices. In this paper, we show that graphs of diameter 2, maximum degree d and d2-1 vertices do not exist for most values of d, when d is even, and we conjecture that they do not exist for any even d greater than 4.
Description:
In the pursuit of obtaining largest graphs of given degree and diameter, many construction techniques have arisen. Compounding of graphs is one such technique. In this paper, by means of the compounding of complete graphs into the bipartite Moore graph of diameter 6, we obtain two families of (
Description:
We consider graphs of maximum degree 3, diameter D≥2 and at most 4 vertices less than the Moore bound M3,D, that is, (3,D,−)-graphs for ≤4. We prove the non-existence of (3,D,−4)-graphs for D≥5, completing in this way the catalogue of (3,D,−)-graphs with D≥2 and ≤4. Our results also give an improvement to the upper bound on the largest possible number N3,D of vertices in a graph of maximum degree 3 and diameter D, so that N3,D≤M3,D−6 for D≥5. Copyright Elsevier.
Description:
We consider graphs of maximum degree 3, diameter D≥2 and at most 4 vertices less than the Moore bound M3,D, that is, (3,D,−)-graphs for ≤4. We prove the non-existence of (3,D,−4)-graphs for D≥5, completing in this way the catalogue of (3,D,−)-graphs with D≥2 and ≤4. Our results also give an improvement to the upper bound on the largest possible number N3,D of vertices in a graph of maximum degree 3 and diameter D, so that N3,D≤M3,D−6 for D≥5. Copyright Elsevier.
Description:
It is well known that apart from the Petersen graph there are no Moore graphs of degree 3. As a cubic graph must have an even number of vertices, there are no graphs of maximum degree 3 and
Description:
We introduce the problem of finding the largest subgraph of a given weighted undirected graph (host graph), subject to constraints on the maximum degree and the diameter. We discuss some applications in security, network design and parallel processing, and in connection with the latter we derive some bounds for the order of the largest subgraph in host graphs of practical interest: the mesh and the hypercube. We also present a heuristic strategy to solve the problem, and we prove an approximation ratio for the algorithm. Finally, we provide some experimental results with a variety of host networks, which show that the algorithm performs better in practice than the prediction provided by our theoretical approximation ratio.
Description:
There is an increasing public and policy awareness that tracking cookies are being used to support behavioral advertising, but the extent to which tracking is occurring is not clear. The extent of tracking could have implications for the enforceability of legislative responses to the sharing of personal data, including the Privacy Act 1988 (Cth). In this paper, we develop a methodology for determining the prevalence of tracking cookies, and report the results for a sample of the 50 most visited sites by Australians. We find that the use of tracking cookies is endemic, but that distinct clusters of tracking can be identified across categories including search, pornography and social networking. The implications of the work in relation to privacy are discussed.
Description:
We define the excess degree \xi (P) of a d-polytope P as 2f1 - df0, where f0 and f1 denote the number of vertices and edges, respectively. This parameter measures how much P deviates from being simple. It turns out that the excess degree of a d-polytope does not take every natural number: the smallest possible values are 0 and d - 2, and the value d - 1 only occurs when d = 3 or 5. On the other hand, for fixed d, the number of values not taken by the excess degree is finite if d is odd, and the number of even values not taken by the excess degree is finite if d is even. The excess degree is then applied in three different settings. First, it is used to show that polytopes with small excess (i.e., \xi (P) < d) have a very particular structure: provided d ot = 5, either there is a unique nonsimple vertex, or every nonsimple vertex has degree d + 1. This implies that such polytopes behave in a similar manner to simple polytopes in terms of Minkowski decomposability: they are either decomposable or pyramidal, and their duals are always indecomposable. Second, we characterize completely the decomposable d-polytopes with 2d + 1 vertices (up to combinatorial equivalence). Third, all pairs (f0, f1), for which there exists a 5-polytope with f0 vertices and f1 edges, are determined.
Description:
We define the excess degree \xi (P) of a d-polytope P as 2f1 - df0, where f0 and f1 denote the number of vertices and edges, respectively. This parameter measures how much P deviates from being simple. It turns out that the excess degree of a d-polytope does not take every natural number: the smallest possible values are 0 and d - 2, and the value d - 1 only occurs when d = 3 or 5. On the other hand, for fixed d, the number of values not taken by the excess degree is finite if d is odd, and the number of even values not taken by the excess degree is finite if d is even. The excess degree is then applied in three different settings. First, it is used to show that polytopes with small excess (i.e., \xi (P) < d) have a very particular structure: provided d ot = 5, either there is a unique nonsimple vertex, or every nonsimple vertex has degree d + 1. This implies that such polytopes behave in a similar manner to simple polytopes in terms of Minkowski decomposability: they are either decomposable or pyramidal, and their duals are always indecomposable. Second, we characterize completely the decomposable d-polytopes with 2d + 1 vertices (up to combinatorial equivalence). Third, all pairs (f0, f1), for which there exists a 5-polytope with f0 vertices and f1 edges, are determined.