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

2020
*
arXiv
*
pre-print

We show that the

arXiv:2007.12819v2
fatcat:tlc73pogandjlkvkbmrdgcn5xa
*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*. ...##
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Non-backtracking Spectrum: Unitary Eigenvalues and Diagonalizability
[article]

2020
*
arXiv
*
pre-print

Much effort has been spent on characterizing the spectrum

arXiv:2007.13611v2
fatcat:ag2z4aywlbde3pgkrywn5jesxu
*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. ...##
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Bounds on the Estrada index of ISR (4,6)-fullerenes

2011
*
Applied Mathematics Letters
*

Suppose G is a

doi:10.1016/j.aml.2010.10.018
fatcat:33hnhrrxdvcpngnfxgghz57e54
*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). ...##
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On the relation between quantum walks and zeta functions
[article]

2011
*
arXiv
*
pre-print

We present an explicit formula for the characteristic polynomial

arXiv:1103.0079v2
fatcat:rvg5gkegp5e65nr7gzfxuhaztu
*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). ...##
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Quantum walks do not like bridges
[article]

2021
*
arXiv
*
pre-print

We achieve this result by applying the 1-sum lemma for the characteristic polynomial

arXiv:2112.03374v1
fatcat:7o5a3y2xffafxow4epk2lselji
*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) ...##
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Trees with Four and Five Distinct Signless Laplacian Eigenvalues

2019
*
Journal of the Indonesian Mathematical Society
*

to $j$ in $G$

doi:10.22342/jims.25.3.557.302-313
fatcat:pr7s2wc4cjh7rmxy2g42b5uwii
*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. ...##
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Spectral bounds for the k-independence number of a graph
[article]

2016
*
arXiv
*
pre-print

We construct

arXiv:1510.07186v2
fatcat:tr6owzftqnd2tbt3sdof7yxbca
*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. ...##
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The spectral approach to determining the number of walks in a graph

1979
*
Pacific Journal of Mathematics
*

An elementary

doi:10.2140/pjm.1979.80.443
fatcat:5yh2g3wc3vfyloopv6wyc3aqum
*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*. ...##
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The second eigenvalue of regular graphs of given girth

1992
*
Journal of combinatorial theory. Series B (Print)
*

Lower bounds on the subdominant

doi:10.1016/0095-8956(92)90020-x
fatcat:ivuiv5l7ivf4vodnek55poz53u
*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. ...##
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Real State Transfer
[article]

2017
*
arXiv
*
pre-print

A continuous quantum

arXiv:1710.04042v1
fatcat:onmvacwpifc5xafmqef532t6jq
*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. ...##
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Spectra of Random Regular Hypergraphs

2021
*
Electronic Journal of Combinatorics
*

We then relate the

doi:10.37236/8741
fatcat:fehq5l7sxzc5rffc2sbaqmqqb4
*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. ...##
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Quantum walks defined by digraphs and generalized Hermitian adjacency matrices
[article]

2019
*
arXiv
*
pre-print

Furthermore, we give definitions

arXiv:1910.12536v1
fatcat:lhobyxpoyrc3xlwvfmtckmw5qu
*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 . ...##
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On the spectral distribution of large weighted random regular graphs
[article]

2013
*
arXiv
*
pre-print

Our analysis uses combinatorial results about

arXiv:1306.6714v1
fatcat:gjjdae2ovzde7dw47jnehvxa4e
*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*). ...##
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Eigenvalue Spacings for Regular Graphs
[chapter]

1999
*
IMA Volumes in Mathematics and its Applications
*

A review

doi:10.1007/978-1-4612-1544-8_12
fatcat:37rbwzhxevhtbdfpaz4bvi6op4
*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. ...##
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On the spectral distribution of large weighted random regular graphs

2014
*
Random Matrices. Theory and Applications
*

Our analysis uses combinatorial results about

doi:10.1142/s2010326314500154
fatcat:3x3rm4qxf5g7xax5ivpxelzcdq
*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*). ...
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