1,790 research outputs found
The number of vertices of degree 5 in a contraction-critically 5-connected graph
AbstractAn edge of a 5-connected graph is said to be 5-contractible if the contraction of the edge results in a 5-connected graph. A 5-connected graph with no 5-contractible edge is said to be contraction-critically 5-connected. Let V(G) and V5(G) denote the vertex set of a graph G and the set of degree 5 vertices of G, respectively. We prove that each contraction-critically 5-connected graph G has at least |V(G)|/2 vertices of degree 5. We also show that there is a sequence of contraction-critically 5-connected graphs {Gi} such that limiāā|V5(Gi)|/|V(Gi)|=1/2
Vertices of degree 5 in a contraction critically 5-connected graph
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Explicit Spectral Decimation for a Class of Self--Similar Fractals
The method of spectral decimation is applied to an infinite collection of
self--similar fractals. The sets considered belong to the class of nested
fractals, and are thus very symmetric. An explicit construction is given to
obtain formulas for the eigenvalues of the Laplace operator acting on these
fractals
Minimal Connectivity
A k-connected graph such that deleting any edge / deleting any vertex /
contracting any edge results in a graph which is not k-connected is called
minimally / critically / contraction-critically k-connected. These three
classes play a prominent role in graph connectivity theory, and we give a brief
introduction with a light emphasis on reduction- and construction theorems for
classes of k-connected graphs.Comment: IMADA-preprint-math, 33 page
A degree and forbidden subgraph condition for a k-contractible edge
An edge in a k-conected graph is said to be k-contractible if the contraction of it results in a k-connected graph. We say that k-connected graph G satis es ā degree-sum conditon āif Ī£x2V (W)degG(x) 3k +2 holds for any connected subgraph W of G with ćļ½Wļ½= 3. Let k be an integer such that k 5. We prove that if a k-connected graph with no K1+C4 satis es degree-sum condition, then it has a k-contractible edge.é»ę°éäæ”大å¦201
Zero-free regions for multivariate Tutte polynomials (alias Potts-model partition functions) of graphs and matroids
The chromatic polynomial P_G(q) of a loopless graph G is known to be nonzero
(with explicitly known sign) on the intervals (-\infty,0), (0,1) and (1,32/27].
Analogous theorems hold for the flow polynomial of bridgeless graphs and for
the characteristic polynomial of loopless matroids. Here we exhibit all these
results as special cases of more general theorems on real zero-free regions of
the multivariate Tutte polynomial Z_G(q,v). The proofs are quite simple, and
employ deletion-contraction together with parallel and series reduction. In
particular, they shed light on the origin of the curious number 32/27.Comment: LaTeX2e, 49 pages, includes 5 Postscript figure
Rearrangement Groups of Fractals
We construct rearrangement groups for edge replacement systems, an infinite
class of groups that generalize Richard Thompson's groups F, T, and V .
Rearrangement groups act by piecewise-defined homeomorphisms on many
self-similar topological spaces, among them the Vicsek fractal and many Julia
sets. We show that every rearrangement group acts properly on a locally finite
CAT(0) cubical complex, and we use this action to prove that certain
rearrangement groups are of type F infinity.Comment: 48 pages, 37 figure
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