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Support of Closed Walks and Second Eigenvalue Multiplicity of the Normalized Adjacency Matrix [article]

Theo McKenzie, Peter M. R. Rasmussen, Nikhil Srivastava
2020 arXiv   pre-print
We show that the multiplicity of the second normalized adjacency matrix eigenvalue of any connected graph of maximum degree Δ is bounded by O(n Δ^7/5/log^1/5-o(1)n) for any Δ, and by O(nlog^1/2d/log^1/  ...  The main ingredient in the proof is a polynomial (in k) lower bound on the typical support of a closed random walk of length 2k in any connected graph, which in turn relies on new lower bounds for the  ...  (from the set of all closed walks) in an irregular graph must have large support.  ... 
arXiv:2007.12819v2 fatcat:tlc73pogandjlkvkbmrdgcn5xa

Non-backtracking Spectrum: Unitary Eigenvalues and Diagonalizability [article]

Leo Torres
2020 arXiv   pre-print
Much effort has been spent on characterizing the spectrum of the non-backtracking matrix of certain classes of graphs, with special emphasis on the leading eigenvalue or the second eigenvector.  ...  We relate the multiplicities of such eigenvalues to the existence of specific subgraphs.  ...  Acknowledgements This work was supported by NSF IIS-1741197.  ... 
arXiv:2007.13611v2 fatcat:ag2z4aywlbde3pgkrywn5jesxu

Bounds on the Estrada index of ISR (4,6)-fullerenes

A.R. Ashrafi, G.H. Fath-Tabar
2011 Applied Mathematics Letters  
Suppose G is a graph and λ 1 , λ 2 , . . . λ n are the eigenvalues of G. The Estrada index EE(G) of G is defined as the sum of the terms e λ i , 1 ≤ i ≤ n.  ...  In this work some upper and lower bounds for the Estrada index of (4, 6)-fullerene graphs are presented.  ...  This research is partially supported by Iran National Science Foundation (INSF) (Grant No. 87041993).  ... 
doi:10.1016/j.aml.2010.10.018 fatcat:33hnhrrxdvcpngnfxgghz57e54

On the relation between quantum walks and zeta functions [article]

Norio Konno, Iwao Sato
2011 arXiv   pre-print
We present an explicit formula for the characteristic polynomial of the transition matrix of the discrete-time quantum walk on a graph via the second weighted zeta function.  ...  As applications, we obtain new proofs for the results on spectra of the transition matrix and its positive support.  ...  The second author was partially supported by the Grant-in-Aid for Scientific Research (C) of Japan Society for the Promotion of Science (Grant No. 19540154).  ... 
arXiv:1103.0079v2 fatcat:rvg5gkegp5e65nr7gzfxuhaztu

Quantum walks do not like bridges [article]

Gabriel Coutinho, Chris Godsil, Emanuel Juliano, Christopher M. van Bommel
2021 arXiv   pre-print
We achieve this result by applying the 1-sum lemma for the characteristic polynomial of graphs, the neutrino identities that relate entries of eigenprojectors and eigenvalues, and variational principles  ...  We see our result as an intermediate step to broaden the understanding of how connectivity plays a key role in quantum walks, and as further evidence of the conjecture that no tree on four or more vertices  ...  Let W a (X; t) be the walk generating function for the closed walks that start and end at vertex a (thus, the coefficient of x k counts the number of closed walks that start and end at a after k steps)  ... 
arXiv:2112.03374v1 fatcat:7o5a3y2xffafxow4epk2lselji

Trees with Four and Five Distinct Signless Laplacian Eigenvalues

Fatemeh Taghvaee, Gholam Hossein Fath-Tabar
2019 Journal of the Indonesian Mathematical Society  
to $j$ in $G$ and $A_{ij} = 0$ otherwise‎.‎The eigenvalues of $Q$ is called the signless Laplacian eigenvalues of $G$ and denoted by $q_1$‎, ‎$q_2$‎, ‎$\cdots$‎, ‎$q_n$ in a graph with $n$ vertices‎.‎In  ...  this paper we characterize all trees with four and five distinct signless Laplacian ‎eigenvalues.‎‎‎  ...  The research of this paper is partially supported by the University of Kashan under grant no 504631/12.  ... 
doi:10.22342/jims.25.3.557.302-313 fatcat:pr7s2wc4cjh7rmxy2g42b5uwii

Spectral bounds for the k-independence number of a graph [article]

Aida Abiad, Sebastian Cioabă, Michael Tait
2016 arXiv   pre-print
We construct graphs that attain equality for our first bound and show that our second bound compares favorably to previous bounds on the k-independence number.  ...  In this paper, we obtain two spectral upper bounds for the k-independence number of a graph which is is the maximum size of a set of vertices at pairwise distance greater than k.  ...  Some of this work was done when the first and third authors were at the SP Coding and Information School in Campinas, Brazil. We gratefully acknowledge support from UNICAMP and the school organizers.  ... 
arXiv:1510.07186v2 fatcat:tr6owzftqnd2tbt3sdof7yxbca

The spectral approach to determining the number of walks in a graph

Frank Harary, Allen Schwenk
1979 Pacific Journal of Mathematics  
An elementary graph theoretic interpretation identifies the trace of A n as the number of closed walks of length n in G.  ...  Thus, the main eigenvalues are the roots of T and the nonmain eigenvalues are readily found from Theorem 4. We conclude by returning to the enumeration of closed walks.  ... 
doi:10.2140/pjm.1979.80.443 fatcat:5yh2g3wc3vfyloopv6wyc3aqum

The second eigenvalue of regular graphs of given girth

Patrick Solé
1992 Journal of combinatorial theory. Series B (Print)  
Lower bounds on the subdominant eigenvalue of regular graphs of given girth are derived.  ...  Then the associated orthogonal polynomials coincide up to a degree equal to half the girth, and their extremal zeroes provide bounds on the supports of these distributions.  ...  Thanks are due to the referees for helpful suggestions which greatly improved the presentation of the material.  ... 
doi:10.1016/0095-8956(92)90020-x fatcat:ivuiv5l7ivf4vodnek55poz53u

Real State Transfer [article]

Chris Godsil
2017 arXiv   pre-print
A continuous quantum walk on a graph X with adjacency matrix A is specified by the 1-parameter family of unitary matrices U(t)=(itA).  ...  As a consequence of these we derive strong restrictions on the occurence of uniform mixing on bipartite graphs and on oriented graphs.  ...  First, S is discrete and consists of all integer multiples of its least positive element. Second, S is dense in R and there is sequence of positive elements (σ i ) i≥0 with limit 0.  ... 
arXiv:1710.04042v1 fatcat:onmvacwpifc5xafmqef532t6jq

Spectra of Random Regular Hypergraphs

Ioana Dumitriu, Yizhe Zhu
2021 Electronic Journal of Combinatorics  
We then relate the second eigenvalues to both its expansion property and the mixing rate of the non-backtracking random walk on regular hypergraphs.  ...  In this paper, we study the spectra of regular hypergraphs following the definitions from Feng and Li (1996).  ...  Acknowledgments We thank Sebastian Cioabȃ and Kameron Decker Harris for helpful comments. This work was partially supported by NSF DMS-1949617.  ... 
doi:10.37236/8741 fatcat:fehq5l7sxzc5rffc2sbaqmqqb4

Quantum walks defined by digraphs and generalized Hermitian adjacency matrices [article]

Sho Kubota, Etsuo Segawa, Tetsuji Taniguchi
2019 arXiv   pre-print
Furthermore, we give definitions of the positive and negative supports of the transfer matrix, and clarify explicit formulas of their supports of the square.  ...  We propose a quantum walk defined by digraphs (mixed graphs). This is like Grover walk that is perturbed by a certain complex-valued function defined by digraphs.  ...  In order to determine the multiplicity of the eigenvalues ±1, we next investigate the value I(c) for any closed path c on Y ± a,n−a .  ... 
arXiv:1910.12536v1 fatcat:lhobyxpoyrc3xlwvfmtckmw5qu

On the spectral distribution of large weighted random regular graphs [article]

Leo Goldmakher, Cap Khoury, Steven J. Miller, Kesinee Ninsuwan
2013 arXiv   pre-print
Our analysis uses combinatorial results about closed acyclic walks in large trees, which may be of independent interest.  ...  McKay proved that the limiting spectral measures of the ensembles of d-regular graphs with N vertices converge to Kesten's measure as N→∞. In this paper we explore the case of weighted graphs.  ...  a closed walk of length k in G (where by the weight of a walk we mean the product of the weights of all edges traversed, counted with multiplicity).  ... 
arXiv:1306.6714v1 fatcat:gjjdae2ovzde7dw47jnehvxa4e

Eigenvalue Spacings for Regular Graphs [chapter]

Dmitry Jakobson, Stephen D. Miller, Igor Rivin, Zeév Rudnick
1999 IMA Volumes in Mathematics and its Applications  
A review of the basic facts on graphs and their spectra is included.  ...  We carry out a numerical study of fluctuations in the spectrum of regular graphs.  ...  Accordingly, the trace of A r is equal to the number of closed walks of length r.  ... 
doi:10.1007/978-1-4612-1544-8_12 fatcat:37rbwzhxevhtbdfpaz4bvi6op4

On the spectral distribution of large weighted random regular graphs

Leo Goldmakher, Cap Khoury, Steven J. Miller, Kesinee Ninsuwan
2014 Random Matrices. Theory and Applications  
Our analysis uses combinatorial results about closed acyclic walks in large trees, which may be of independent interest. 2010 Mathematics Subject Classification. 15B52, 05C80, 60F05 (primary), 05C22, 05C38  ...  McKay proved the limiting spectral measures of the ensembles of d-regular graphs with N vertices converge to Kesten's measure as N → ∞.  ...  a closed walk of length k in G (where by the weight of a walk we mean the product of the weights of all edges traversed, counted with multiplicity).  ... 
doi:10.1142/s2010326314500154 fatcat:3x3rm4qxf5g7xax5ivpxelzcdq
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