scholarly journals A Spectral Characterization for Concentration of the Cover Time

2019 ◽  
Vol 33 (4) ◽  
pp. 2167-2184
Author(s):  
Jonathan Hermon
Keyword(s):  
Markov Chains ◽  
Spectral Gap ◽  
Hitting Time ◽  
Spectral Characterization ◽  
Cover Time ◽  
Cover Times ◽  
Reversible Markov Chains ◽  
Vertex Transitive ◽  
Vertex Transitive Graphs ◽  
Transitive Graphs

Abstract We prove that for a sequence of finite vertex-transitive graphs of increasing sizes, the cover times are asymptotically concentrated if and only if the product of the spectral gap and the expected cover time diverges. In fact, we prove this for general reversible Markov chains under the much weaker assumption (than transitivity) that the maximal hitting time of a state is of the same order as the average hitting time.

Trivalent Non-symmetric Vertex-Transitive Graphs of Order at Most 150

2008 ◽  
Vol 15 (03) ◽  
pp. 379-390 ◽  
Author(s):  
Xuesong Ma ◽  
Ruji Wang
Keyword(s):  
Automorphism Group ◽  
Bipartite Graphs ◽  
Type Ii ◽  
Symmetric Graph ◽  
Trivalent Graph ◽  
Block Graphs ◽  
Group A ◽  
Vertex Transitive ◽  
Vertex Transitive Graphs ◽  
Transitive Graphs

Let X be a simple undirected connected trivalent graph. Then X is said to be a trivalent non-symmetric graph of type (II) if its automorphism group A = Aut (X) acts transitively on the vertices and the vertex-stabilizer Av of any vertex v has two orbits on the neighborhood of v. In this paper, such graphs of order at most 150 with the basic cycles of prime length are investigated, and a classification is given for such graphs which are non-Cayley graphs, whose block graphs induced by the basic cycles are non-bipartite graphs.

Properties of Classes of Random Graphs

1994 ◽  
Vol 3 (4) ◽  
pp. 435-454 ◽  
Author(s):  
Neal Brand ◽  
Steve Jackson
Keyword(s):  
Random Graphs ◽  
Higher Order ◽  
Order Property ◽  
First Order ◽  
New Class ◽  
Vertex Transitive ◽  
Vertex Transitive Graphs ◽  
Almost All ◽  
Transitive Graphs

In [11] it is shown that the theory of almost all graphs is first-order complete. Furthermore, in [3] a collection of first-order axioms are given from which any first-order property or its negation can be deduced. Here we show that almost all Steinhaus graphs satisfy the axioms of almost all graphs and conclude that a first-order property is true for almost all graphs if and only if it is true for almost all Steinhaus graphs. We also show that certain classes of subgraphs of vertex transitive graphs are first-order complete. Finally, we give a new class of higher-order axioms from which it follows that large subgraphs of specified type exist in almost all graphs.

scholarly journals Cubic vertex-transitive graphs on up to 1280 vertices

2013 ◽  
Vol 50 ◽  
pp. 465-477 ◽  
Author(s):  
Primož Potočnik ◽  
Pablo Spiga ◽  
Gabriel Verret
Keyword(s):  
Vertex Transitive ◽  
Vertex Transitive Graphs ◽  
Transitive Graphs
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