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## Filters

##
###
Inequalities for the Number of Walks in Graphs

2013
*
Algorithmica
*

We investigate

doi:10.1007/s00453-013-9766-3
fatcat:5nmq7q2axrcr7dlwgfjpqm4emm
*the*growth*of**the**number*w k*of**walks**of*length k*in*undirected*graphs*as well as related*inequalities*. ...*In**the*first part, we derive*the**inequalities*w 2a+c · w 2(a+b)+c ≤ w 2a · w 2(a+b+c) and w 2a+c ... Acknowledgments We want to thank Daniel Fleischer, Alexander Offtermatt-Souza, Moritz Maaß, Riko Jacob, and Holger Täubig*for*valuable remarks and discussions. ...##
###
Upper Bounds For Hitting Times Of Random Walks On Sparse Graphs
[article]

2017
*
arXiv
*
pre-print

We obtain upper bounds (

arXiv:1702.04026v1
fatcat:3fqyhrir7jb2honlnq7f2hnodq
*in*most cases, sharp)*for**the*hitting times*of*random*walks*on finite undirected*graphs*expressed as functions*of**the*graph's*number**of*edges. ...*In*particular, we show that*the*maximum hitting time*for*a simple random*walk*on a connected*graph*with m edges is at most m^2. ... Thus proving our*inequality**for*′ will prove it*for**the*original*graph*, because both*number**of*edges and have not increased when we switched from to ′ . ...##
###
The Constrained Crossing Minimization Problem
[chapter]

1999
*
Lecture Notes in Computer Science
*

*number*

*of*crossings between

*the*

*walks*. ... Here we present an integer linear programming formulation (ILP)

*for*

*the*shortest crossing

*walks*problem. Furthermore, we will present additional valid

*inequalities*that strengthen

*the*formulation. ...

*The*basic idea

*for*representing a set

*of*

*walks*

*in*terms

*of*linear

*inequalities*is to use variables

*for*pairs

*of*adjacent edges instead

*of*variables

*for*

*the*edges only. ...

##
###
On the Laplacian-energy-like invariant

2014
*
Linear Algebra and its Applications
*

*In*this paper, some upper and lower bounds

*for*LEL, as well as, some lower bounds

*for*

*the*spectral radius

*of*

*graph*are obtained. ...

*The*Laplacian-energy-like

*of*a simple connected

*graph*is defined as LEL = LEL(G) = ∑ , where ( ) ≥ ( ) ≥ ⋯ ≥ ( ) = 0 are

*the*Laplacian eigenvalues

*of*

*the*

*graph*. ...

*The*

*number*

*of*closed

*walks*

*in*

*of*length ℓ starting at is thus given by ( ( ) ℓ ) , so

*the*total

*number*(ℓ)

*of*closed

*walks*

*of*length ℓ is given by (ℓ) = ∑ ( ( ) ℓ ) = tr( ( ) ℓ ) where tr denotes

*the*trace ...

##
###
New bounds for the max- k -cut and chromatic number of a graph

2016
*
Linear Algebra and its Applications
*

*For*regular

*graphs*,

*the*new bound on

*the*chromatic

*number*is

*the*same as

*the*well-known Hoffman bound; however,

*the*two bounds are incomparable

*in*general. ... We investigate

*the*presented bounds

*for*specific classes

*of*

*graphs*, such as

*walk*-regular

*graphs*, strongly regular

*graphs*, and

*graphs*from

*the*Hamming association scheme. ...

*The*matrix A contains

*the*

*numbers*

*of*

*walks*

*of*length between vertices, so

*the*definition is equivalent to requiring that

*the*

*number*

*of*

*walks*

*of*length from a vertex to itself is

*the*same

*for*every vertex ...

##
###
Lower bounds for the Estrada index using mixing time and Laplacian spectrum

2013
*
Rocky Mountain Journal of Mathematics
*

We derive novel analytic lower bounds

doi:10.1216/rmj-2013-43-6-2009
fatcat:arz4xo6b7vbqra3czjl3sq4z5m
*for**the*logarithm*of**the*Estrada index based on*the*Laplacian spectrum and*the*mixing times*of*random*walks*on*the*network. ...*The*main techniques employed are some*inequalities*, such as*the*thermodynamic*inequality**in*statistical mechanics, a trace*inequality**of*von Neumann, and a refined harmonic-arithmetic mean*inequality*. ... A weighted sum*of**numbers**of*closed*walks*is defined*in*[28] by S = ∞ k=0 n k /k!, where n k is*the**number**of*closed*walks**of*length k*in*G. ...##
###
A note on a walk-based inequality for the index of a signed graph

2021
*
Special Matrices
*

AbstractWe derive an

doi:10.1515/spma-2020-0120
fatcat:nd7usezxdjeyhas6x6tdjhnlsi
*inequality*that includes*the*largest eigenvalue*of**the*adjacency matrix and*walks**of*an arbitrary length*of*a signed*graph*. We also consider certain particular cases. ... Acknowledgements: Research is partially supported by Serbian Ministry*of*Education via Faculty*of*Mathematics, University*of*Belgrade. ... Data Availability Statement: Data sharing is not applicable to this article as no datasets were generated or analyzed during*the*current study. ...##
###
Bounds on the number of closed walks in a graph and its applications

2014
*
Journal of Inequalities and Applications
*

This

doi:10.1186/1029-242x-2014-199
fatcat:vinozvufcjemle774c4jdm442a
*inequality*yields several upper bounds*for**the**number**of*closed*walks**in*a*graph**in*terms*of**the**number**of*vertices,*number**of*edges, maximum degree, degree sequence, and*the*Zagreb indices*of**the*... Using*graph*-theoretical techniques, we establish an*inequality*regarding*the**number**of**walks*and closed*walks**in*a*graph*. ... Acknowledgements*The*authors would like to thank*the*anonymous referees*for*their extremely helpful comments and suggestions towards improving*the*original version*of*this paper. ...##
###
On a poset of trees

2010
*
Combinatorica
*

We will prove that

doi:10.1007/s00493-010-2516-0
fatcat:ne3djct47zbonp33ejffxivr6u
*the*path minimizes*the**number**of*closed*walks**of*length ℓ among*the*connected*graphs**for*all ℓ. ... Indeed, we will prove that*the**number**of*closed*walks**of*length ℓ and many other properties such as*the*spectral radius, Estada index increase or decrease along a certain poset*of*trees. ...*The*author is very grateful to Vladimir Nikiforov and László Lovász*for*their numerous advice and comments. ...##
###
Inequalities for Laplacian Eigenvalues of Signed Graphs with Given Frustration Number

2021
*
Symmetry
*

*The*frustration

*number*f

*of*a signed

*graph*is

*the*size

*of*

*the*minimal set F

*of*vertices whose removal results

*in*a balanced signed

*graph*; hence, a connected signed

*graph*G˙ is balanced if and only if f ...

*In*this paper, we consider

*the*balance

*of*G˙ via

*the*relationships between

*the*frustration

*number*and eigenvalues

*of*

*the*symmetric Laplacian matrix associated with G˙. ... By

*the*choice

*of*l, we have that

*the*

*number*

*of*

*walks*

*of*

*the*first type

*in*G is equal to

*the*difference

*of*positive and negative

*walks*

*of*

*the*same type inĠ. ...

##
###
Walks and the spectral radius of graphs

2006
*
Linear Algebra and its Applications
*

Given a

doi:10.1016/j.laa.2006.02.003
fatcat:jqym7vq6jjg57jh2ee2tc2z6gm
*graph*G, write µ(G)*for**the*largest eigenvalue*of*its adjacency matrix, ω(G)*for*its clique*number*, and w k (G)*for**the**number**of*its k-*walks*. ... We prove that*the**inequalities*hold*for*all r > 0 and odd q > 0. We also generalize a*number**of*other bounds on µ(G) and characterize semiregular and pseudo-regular*graphs**in*spectral terms. ... Acknowledgments I am grateful to*the*referee*for*his valuable remarks. ...##
###
Number of walks and degree powers in a graph

2009
*
Discrete Mathematics
*

This note deals with

doi:10.1016/j.disc.2008.03.025
fatcat:4ufcojpel5hr7iao3s2blozgky
*the*relationship between*the*total*number**of*k-*walks**in*a*graph*, and*the*sum*of**the*k-th powers*of*its vertex degrees. ...*In*particular, it is shown that*the**the**number**of*all k-*walks*is upper bounded by*the*sum*of**the*k-th powers*of**the*degrees. ...*The*authors thank one*of**the*referees*for**the*comment about*the*use*of**the*Hölder*inequality*and*the*result*in*(2) . ...##
###
Walks and the spectral radius of graphs
[article]

2006
*
arXiv
*
pre-print

We give upper and lower bounds on

arXiv:math/0506259v2
fatcat:ojl5ebm44ffandynr333jo76ki
*the*spectral radius*of*a*graph**in*terms*of**the**number**of**walks*. We generalize a*number**of*known results. ... Given a*graph*G, a k-*walk*is a sequence*of*vertices v 1 , ..., v k*of*G such that v i is adjacent to v i+1*for*all i = 1, ..., k − 1; we write w k (G)*for**the**number**of*k-*walks**in*G. ... Write cw k (G)*for**the**number**of*closed*walks*on k vertices*in*G (i.e., k-*walks*with*the*same start and end vertex.) ...##
###
Eigenvalues of Graphs
[chapter]

1995
*
Proceedings of the International Congress of Mathematicians
*

*For*any

*graph*H on s vertices,

*the*

*number*

*of*occurrences

*of*H as an induced subgraph

*of*G is (1 + o(1)) times

*the*expected

*number*. ...

*The*situation

*for*

*the*Harnack

*inequalities*

*for*

*graphs*is somewhat different since discrete versions

*of*

*the*statement

*for*

*the*continuous cases do not hold

*in*general. ...

##
###
Maximum walk entropy implies walk regularity

2014
*
Linear Algebra and its Applications
*

*The*notion

*of*

*walk*entropy ... characterization

*of*

*the*

*walk*-regularity

*in*

*graphs*and also gives strong mathematical support to

*the*strength

*of*this

*graph*invariant

*for*studying

*the*structure

*of*

*graphs*and networks. ... Q.E.D. closing,

*the*maximum

*of*

*the*

*walk*entropy at 1 , i.e.,

*for*

*the*

*walk*-regular

*graphs*. ...

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