Invariant random perfect matchings in Cayley graphs

2017 ◽  
Vol 11 (1) ◽  
pp. 211-243 ◽  
Author(s):  
Endre Csóka ◽  
Gabor Lippner
Keyword(s):  
Cayley Graphs ◽  
Perfect Matchings

MATCHING PRECLUSION FOR ALTERNATING GROUP GRAPHS AND THEIR GENERALIZATIONS

2008 ◽  
Vol 19 (06) ◽  
pp. 1413-1437 ◽  
Author(s):  
EDDIE CHENG ◽  
LINDA LESNIAK ◽  
MARC J. LIPMAN ◽  
LÁSZLÓ LIPTÁK
Keyword(s):  
Cayley Graphs ◽  
Perfect Matchings ◽  
Alternating Group ◽  
Optimal Solutions ◽  
Minimum Number

The matching preclusion number of a graph is the minimum number of edges whose deletion results in a graph that has neither perfect matchings nor almost-perfect matchings. In this paper, we find this number for the alternating group graphs, Cayley graphs generated by 2-trees and the (n,k)-arrangement graphs. Moreover, we classify all the optimal solutions.

scholarly journals A Conjecture of Norine and Thomas for Abelian Cayley Graphs

10.37236/5456 ◽  
2017 ◽  
Vol 24 (3) ◽  
Author(s):  
Fuliang Lu ◽  
Lianzhu Zhang
Keyword(s):  
Polynomial Time ◽  
Perfect Matching ◽  
Abelian Groups ◽  
Cayley Graphs ◽  
Perfect Matchings ◽  
Pfaffian Graphs

A graph $\Gamma_1$ is a matching minor of $\Gamma$ if some even subdivision of $\Gamma_1$ is isomorphic to a subgraph $\Gamma_2$ of $\Gamma$, and by deleting the vertices of $\Gamma_2$ from $\Gamma$ the left subgraph has a perfect matching. Motivated by the study of Pfaffian graphs (the numbers of perfect matchings of these graphs can be computed in polynomial time), we characterized Abelian Cayley graphs which do not contain a $K_{3,3}$ matching minor. Furthermore, the Pfaffian property of Cayley graphs on Abelian groups is completely characterized. This result confirms that the conjecture posed by Norine and Thomas in 2008 for Abelian Cayley graphs is true.

scholarly journals Cayley Graphs Without a Bounded Eigenbasis

2020 ◽  
Author(s):  
Ashwin Sah ◽  
Mehtaab Sawhney ◽  
Yufei Zhao
Keyword(s):  
Cayley Graph ◽  
Orthonormal Basis ◽  
Cayley Graphs ◽  
The Other ◽  
Transitive Graph ◽  
Classic Result ◽  
Other Hand ◽  
Small Set ◽  
Vertex Transitive ◽  
Vertex Transitive Graph

Abstract Does every $n$-vertex Cayley graph have an orthonormal eigenbasis all of whose coordinates are $O(1/\sqrt{n})$? While the answer is yes for abelian groups, we show that it is no in general. On the other hand, we show that every $n$-vertex Cayley graph (and more generally, vertex-transitive graph) has an orthonormal basis whose coordinates are all $O(\sqrt{\log n / n})$, and that this bound is nearly best possible. Our investigation is motivated by a question of Assaf Naor, who proved that random abelian Cayley graphs are small-set expanders, extending a classic result of Alon–Roichman. His proof relies on the existence of a bounded eigenbasis for abelian Cayley graphs, which we now know cannot hold for general groups. On the other hand, we navigate around this obstruction and extend Naor’s result to nonabelian groups.

scholarly journals Tetravalent edge-transitive Cayley graphs of Frobenius groups

2021 ◽  
Vol 53 (2) ◽  
pp. 527-551
Author(s):  
Lei Wang ◽  
Yin Liu ◽  
Yanxiong Yan
Keyword(s):  
Cayley Graphs ◽  
Frobenius Groups ◽  
Edge Transitive
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