68 research outputs found
Edge-disjoint spanning trees and eigenvalues of regular graphs
Partially answering a question of Paul Seymour, we obtain a sufficient
eigenvalue condition for the existence of edge-disjoint spanning trees in a
regular graph, when . More precisely, we show that if the second
largest eigenvalue of a -regular graph is less than
, then contains at least edge-disjoint spanning
trees, when . We construct examples of graphs that show our
bounds are essentially best possible. We conjecture that the above statement is
true for any .Comment: 4 figure
The spectrum and toughness of regular graphs
In 1995, Brouwer proved that the toughness of a connected -regular graph
is at least , where is the maximum absolute value of
the non-trivial eigenvalues of . Brouwer conjectured that one can improve
this lower bound to and that many graphs (especially graphs
attaining equality in the Hoffman ratio bound for the independence number) have
toughness equal to . In this paper, we improve Brouwer's spectral
bound when the toughness is small and we determine the exact value of the
toughness for many strongly regular graphs attaining equality in the Hoffman
ratio bound such as Lattice graphs, Triangular graphs, complements of
Triangular graphs and complements of point-graphs of generalized quadrangles.
For all these graphs with the exception of the Petersen graph, we confirm
Brouwer's intuition by showing that the toughness equals ,
where is the smallest eigenvalue of the adjacency matrix of the
graph.Comment: 15 pages, 1 figure, accepted to Discrete Applied Mathematics, special
issue dedicated to the "Applications of Graph Spectra in Computer Science"
Conference, Centre de Recerca Matematica (CRM), Bellaterra, Barcelona, June
16-20, 201
Mixing Rates of Random Walks with Little Backtracking
Many regular graphs admit a natural partition of their edge set into cliques
of the same order such that each vertex is contained in the same number of
cliques. In this paper, we study the mixing rate of certain random walks on
such graphs and we generalize previous results of Alon, Benjamini, Lubetzky and
Sodin regarding the mixing rates of non-backtracking random walks on regular
graphs.Comment: 31 pages; to appear in the CRM Proceedings Series, published by the
American Mathematical Society as part of the Contemporary Mathematics Serie
On the Spectrum of Wenger Graphs
Let , where is a prime and is an integer. For ,
let and be two copies of the -dimensional vector spaces over the
finite field . Consider the bipartite graph with partite
sets and defined as follows: a point is adjacent to a line if and only if the
following equalities hold: for . We call the graphs Wenger graphs. In this paper, we determine all
distinct eigenvalues of the adjacency matrix of and their
multiplicities. We also survey results on Wenger graphs.Comment: 9 pages; accepted for publication to J. Combin. Theory, Series
A graph partition problem
Given a graph on vertices, for which is it possible to partition
the edge set of the -fold complete graph into copies of ? We show
that there is an integer , which we call the \emph{partition modulus of
}, such that the set of values of for which such a partition
exists consists of all but finitely many multiples of . Trivial
divisibility conditions derived from give an integer which divides
; we call the quotient the \emph{partition index of }. It
seems that most graphs have partition index equal to , but we give two
infinite families of graphs for which this is not true. We also compute
for various graphs, and outline some connections between our problem and the
existence of designs of various types
Eigenvalues and edge-connectivity of regular graphs
AbstractIn this paper, we show that if the second largest eigenvalue of a d-regular graph is less than d-2(k-1)d+1, then the graph is k-edge-connected. When k is 2 or 3, we prove stronger results. Let ρ(d) denote the largest root of x3-(d-3)x2-(3d-2)x-2=0. We show that if the second largest eigenvalue of a d-regular graph G is less than ρ(d), then G is 2-edge-connected and we prove that if the second largest eigenvalue of G is less than d-3+(d+3)2-162, then G is 3-edge-connected
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