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On Minimum Order of Odd Regular Graphs Without Perfect Matching

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 Added by Saptarshi Bej
 Publication date 2014
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and research's language is English




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In this article we have derived the minimum order of an odd regular graph such that the graph has no matching. We have observed that how it is different from the case of even regular graphs. We have checked the consistency of the derived result with Petersens theorem.



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332 - M. Abdi , E. Ghorbani , W. Imrich 2019
Aldous and Fill conjectured that the maximum relaxation time for the random walk on a connected regular graph with $n$ vertices is $(1+o(1)) frac{3n^2}{2pi^2}$. This conjecture can be rephrased in terms of the spectral gap as follows: the spectral gap (algebraic connectivity) of a connected $k$-regular graph on $n$ vertices is at least $(1+o(1))frac{2kpi^2}{3n^2}$, and the bound is attained for at least one value of $k$. Based upon previous work of Brand, Guiduli, and Imrich, we prove this conjecture for cubic graphs. We also investigate the structure of quartic (i.e. 4-regular) graphs with the minimum spectral gap among all connected quartic graphs. We show that they must have a path-like structure built from specific blocks.
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Let $X$ be a connected Cayley graph on an abelian group of odd order, such that no two distinct vertices of $X$ have exactly the same neighbours. We show that the direct product $X times K_2$ (also called the canonical double cover of $X$) has only the obvious automorphisms (namely, the ones that come from automorphisms of its factors $X$ and $K_2$). This means that $X$ is stable. The proof is short and elementary. The theory of direct products implies that $K_2$ can be replaced with members of a much more general family of connected graphs.
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