6,068 research outputs found
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
Minimizing the number of independent sets in triangle-free regular graphs
Recently, Davies, Jenssen, Perkins, and Roberts gave a very nice proof of the
result (due, in various parts, to Kahn, Galvin-Tetali, and Zhao) that the
independence polynomial of a -regular graph is maximized by disjoint copies
of . Their proof uses linear programming bounds on the distribution of
a cleverly chosen random variable. In this paper, we use this method to give
lower bounds on the independence polynomial of regular graphs. We also give new
bounds on the number of independent sets in triangle-free regular graphs
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
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