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Register a new userLimbs and Cospectral Vertices in Trees
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We generalize Schwenks result that almost all trees contain any given limb to trees with positive integer vertex weights. The concept of characteristic polynomial is extended to such weighted trees and we prove almost all weighted trees have a cospectral mate. We also prove almost all trees contain $k$ cospectral vertices for any integer $kge2$.
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We prove an upper bound on the number of pairwise strongly cospectral vertices in a normal Cayley graph, in terms of the multiplicities of its eigenvalues. We use this to determine an explicit bound in Cayley graphs of $mathbb{Z}_2^d$ and $mathbb{Z}_4^d$. We also provide some infinite families of Cayley graphs of $mathbb{Z}_2^d$ with a set of four pairwise strongly cospectral vertices and show that such graphs exist in every dimension.
In this paper we enumerate and give bijections for the following four sets of vertices among rooted ordered trees of a fixed size: (i) first-children of degree $k$ at level $ell$, (ii) non-first-children of degree $k$ at level $ell-1$, (iii) leaves having $k-1$ elder siblings at level $ell$, and (iv) non-leaves of outdegree $k$ at level $ell-1$. Our results unite and generalize several previous works in the literature.
In this paper we enumerate the cardinalities for the set of all vertices of outdegree $ge k$ at level $ge ell$ among all rooted ordered $d$-trees with $n$ edges. Our results unite and generalize several previous works in the literature.
Let $G$ be an $n$-vertex graph with adjacency matrix $A$, and $W=[e,Ae,ldots,A^{n-1}e]$ be the walk matrix of $G$, where $e$ is the all-one vector. In Wang [J. Combin. Theory, Ser. B, 122 (2017): 438-451], the author showed that any graph $G$ is uniquely determined by its generalized spectrum (DGS) whenever $2^{-lfloor n/2 rfloor}det W$ is odd and square-free. In this paper, we introduce a large family of graphs $mathcal{F}_n={$ $n$-vertex graphs $Gcolon, 2^{-lfloor n/2 rfloor}det W =p^2b$ and rank$W=n-1$ over $mathbb{Z}/pmathbb{Z}},$ where $b$ is odd and square-free, $p$ is an odd prime and $p mid b$. We prove that any graph in $mathcal{F}_n$ either is DGS or has exactly one generalized cospectral mate up to isomorphism. Moreover, we show that the problem of finding the generalized cospectral mate for a graph in $mathcal{F}_n$ is equivalent to that of generating an appropriate rational orthogonal matrix from a given integral vector. This equivalence essentially depends on an amazing property of graphs in terms of generalized spectra, which states that any symmetric integral matrix generalized cospectral with the adjacency matrix of some graph must be an adjacency matrix. Based on this equivalence, we develop an efficient algorithm to decide whether a given graph in $mathcal{F}_n$ is DGS and further to find the unique generalized cospectral mate when it is not. We give some experimental results on graphs with at most 20 vertices, which suggest that $mathcal{F}_n$ may have a positive density (nearly $3%$) and possibly almost all graphs in $mathcal{F}_n$ are DGS as $nrightarrow infty$. This gives a supporting evidence for Haemers conjecture that almost all graphs are determined by their spectra.
We study regular graphs in which the random walks starting from a positive fraction of vertices have small mixing time. We prove that any such graph is virtually an expander and has no small separator. This answers a question of Pak [SODA, 2002]. As a corollary, it shows that sparse (constant degree) regular graphs with many well-mixing vertices have a long cycle, improving a result of Pak. Furthermore, such cycle can be found in polynomial time. Secondly, we show that if the random walks from a positive fraction of vertices are well-mixing, then the random walks from almost all vertices are well-mixing (with a slightly worse mixing time).
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