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Inequalities for the Number of Walks in Graphs

Hanjo Täubig, Jeremias Weihmann, Sven Kosub, Raymond Hemmecke, Ernst W. Mayr
2013 Algorithmica  
We investigate 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.  ... 
doi:10.1007/s00453-013-9766-3 fatcat:5nmq7q2axrcr7dlwgfjpqm4emm

Upper Bounds For Hitting Times Of Random Walks On Sparse Graphs [article]

Dmitri Fomin
2017 arXiv   pre-print
We obtain upper bounds (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 ′ .  ... 
arXiv:1702.04026v1 fatcat:3fqyhrir7jb2honlnq7f2hnodq

The Constrained Crossing Minimization Problem [chapter]

Petra Mutzel, Thomas Ziegler
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.  ... 
doi:10.1007/3-540-46648-7_18 fatcat:buqfpiuwozfw7ehzfwewu7jogi

On the Laplacian-energy-like invariant

Kinkar Ch. Das, Ivan Gutman, A. Sinan Çevik
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  ... 
doi:10.1016/j.laa.2013.05.002 fatcat:ybrwk5e7e5e63icrdtkqxvmvju

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

E.R. van Dam, R. Sotirov
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  ... 
doi:10.1016/j.laa.2015.09.043 fatcat:u3krjhylc5fdlpi7kxybabqgyi

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

Yilun Shang
2013 Rocky Mountain Journal of Mathematics  
We derive novel analytic lower bounds 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.  ... 
doi:10.1216/rmj-2013-43-6-2009 fatcat:arz4xo6b7vbqra3czjl3sq4z5m

A note on a walk-based inequality for the index of a signed graph

Zoran Stanić
2021 Special Matrices  
AbstractWe derive an 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.  ... 
doi:10.1515/spma-2020-0120 fatcat:nd7usezxdjeyhas6x6tdjhnlsi

Bounds on the number of closed walks in a graph and its applications

Xiaodan Chen, Jianguo Qian
2014 Journal of Inequalities and Applications  
This 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.  ... 
doi:10.1186/1029-242x-2014-199 fatcat:vinozvufcjemle774c4jdm442a

On a poset of trees

Péter Csikvári
2010 Combinatorica  
We will prove that 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.  ... 
doi:10.1007/s00493-010-2516-0 fatcat:ne3djct47zbonp33ejffxivr6u

Inequalities for Laplacian Eigenvalues of Signed Graphs with Given Frustration Number

Milica Anđelić, Tamara Koledin, Zoran Stanić
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.  ... 
doi:10.3390/sym13101902 fatcat:xh35iommbzhcnjmf3kaphssn2q

Walks and the spectral radius of graphs

Vladimir Nikiforov
2006 Linear Algebra and its Applications  
Given a 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.  ... 
doi:10.1016/j.laa.2006.02.003 fatcat:jqym7vq6jjg57jh2ee2tc2z6gm

Number of walks and degree powers in a graph

M.A. Fiol, E. Garriga
2009 Discrete Mathematics  
This note deals with 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) .  ... 
doi:10.1016/j.disc.2008.03.025 fatcat:4ufcojpel5hr7iao3s2blozgky

Walks and the spectral radius of graphs [article]

Vladimir Nikiforov
2006 arXiv   pre-print
We give upper and lower bounds on 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.)  ... 
arXiv:math/0506259v2 fatcat:ojl5ebm44ffandynr333jo76ki

Eigenvalues of Graphs [chapter]

Fan R. K. Chung
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.  ... 
doi:10.1007/978-3-0348-9078-6_128 fatcat:lxmpp4ilgvdmxp5l44lwk63zba

Maximum walk entropy implies walk regularity

Ernesto Estrada, José A. de la Peña
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.  ... 
doi:10.1016/j.laa.2014.06.030 fatcat:qmdcumvgnzgqfmduhdgykzjb2e
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