New inequalities for network distance measures by using graph spectra

Abstract Eigenvalues of graph-theoretical matrices often reflect structure properties of networks (graphs) meaningfully. In this paper, we explore inequalities for graph distance measures which are based on topological indices. Some of these indices are based on eigenvalues of graph-theoretical matrices. We here consider the adjacency matrix, the Laplacian matrix and signless Laplacian matrix. Besides proving the inequalities, we discuss the usefulness of these measures and state some conjectures.

[1]  I. Gutman,et al.  Note on the HOMO-LUMO Index of Graphs ∗ , 2013 .

[2]  Marc Demange,et al.  On some applications of the selective graph coloring problem , 2015, Eur. J. Oper. Res..

[3]  Zengqiang Chen,et al.  SIS model of epidemic spreading on dynamical networks with community , 2009, Frontiers of Computer Science in China.

[4]  Dragoš Cvetković,et al.  Towards a spectral theory of graphs based on the signless Laplacian, III , 2010 .

[5]  Yongtang Shi,et al.  The Maximal Matching Energy of Tricyclic Graphs , 2014, 1409.2038.

[6]  Bin Jiang,et al.  A Structural Approach to the Model Generalization of an Urban Street Network* , 2004, GeoInformatica.

[7]  W. Haemers,et al.  Which graphs are determined by their spectrum , 2003 .

[8]  Igor I. Baskin,et al.  Molecular Similarity. 1. Analytical Description of the Set of Graph Similarity Measures , 1998, J. Chem. Inf. Comput. Sci..

[9]  Yongtang Shi,et al.  Extremality of degree-based graph entropies , 2014, Inf. Sci..

[10]  D. Cvetkovic,et al.  Towards a spectral theory of graphs based on the signless Laplacian, I , 2009 .

[11]  Weijun Liu,et al.  The spectral radius of edge chromatic critical graphs , 2016 .

[12]  Matthias Dehmer,et al.  Strukturelle Analyse web-basierter Dokumente , 2005 .

[13]  Charles R. Johnson,et al.  Matrix analysis , 1985, Statistical Inference for Engineers and Data Scientists.

[14]  Jens Clausen,et al.  A Family of Bipartite Cardinality Matching Problems Solvable in O(n2) Time , 1995, Nordic Journal of Computing.

[15]  Daniel A. Spielman,et al.  Spectral Graph Theory and its Applications , 2007, 48th Annual IEEE Symposium on Foundations of Computer Science (FOCS'07).

[16]  Yongtang Shi,et al.  Interrelations of Graph Distance Measures Based on Topological Indices , 2014, PloS one.

[17]  André Carlos Ponce de Leon Ferreira de Carvalho,et al.  Spectral methods for graph clustering - A survey , 2011, Eur. J. Oper. Res..

[18]  Shuai Ding,et al.  Evolution of cooperation in the traveler's dilemma game on two coupled lattices , 2014, Appl. Math. Comput..

[19]  Yongtang Shi,et al.  On Wiener polarity index of bicyclic networks , 2016, Scientific Reports.

[20]  D. Bozkurt,et al.  On Incidence Energy , 2014 .

[21]  Lihua Feng,et al.  The inertia of weighted unicyclic graphs , 2013, 1307.0059.

[22]  Roberto Todeschini,et al.  Handbook of Molecular Descriptors , 2002 .

[23]  William W. Hager,et al.  Continuous quadratic programming formulations of optimization problems on graphs , 2015, Eur. J. Oper. Res..

[24]  Yongtang Shi,et al.  Note on two generalizations of the Randić index , 2015, Appl. Math. Comput..

[25]  G. Zocchi,et al.  Local cooperativity mechanism in the DNA melting transition. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

[26]  Lihua Feng,et al.  THE SIGNLESS LAPLACIAN SPECTRAL RADIUS OF BOUNDED DEGREE GRAPHS ON SURFACES , 2015 .

[27]  Yongtang Shi,et al.  Fifty years of graph matching, network alignment and network comparison , 2016, Inf. Sci..

[28]  Dragoš Cvetković,et al.  EIGENVALUE BOUNDS FOR THE SIGNLESS LAPLACIAN , 2007 .

[29]  Matthias Dehmer,et al.  Generalized graph entropies , 2011, Complex..

[30]  Thorsten Joachims,et al.  Learning to classify text using support vector machines - methods, theory and algorithms , 2002, The Kluwer international series in engineering and computer science.

[31]  Jakob Krarup,et al.  LP formulations of the shortest path tree problem , 2004, 4OR.

[32]  Yongtang Shi,et al.  A Note on Distance-based Graph Entropies , 2014, Entropy.

[33]  Dragoš Cvetković,et al.  The algebraic multiplicity of the number zero in the spectrum of a bipartite graph , 1972 .

[34]  Yongtang Shi,et al.  Graph distance measures based on topological indices revisited , 2015, Appl. Math. Comput..

[35]  P. Van Mieghem,et al.  Virus Spread in Networks , 2009, IEEE/ACM Transactions on Networking.

[36]  Christos Faloutsos,et al.  Epidemic spreading in real networks: an eigenvalue viewpoint , 2003, 22nd International Symposium on Reliable Distributed Systems, 2003. Proceedings..

[37]  Fred Sobik,et al.  Operations on cognitive structures — their modeling on the basis of graph theory , 1994 .

[38]  Lihua Feng,et al.  The distance spectral radius of graphs with given independence number , 2015, Ars Comb..

[39]  Steven R. Bishop,et al.  Static search games played over graphs and general metric spaces , 2013, Eur. J. Oper. Res..

[40]  Guihai Yu,et al.  Bicyclic graphs with small positive index of inertia , 2013 .

[41]  Chengyi Xia,et al.  An SIR model with infection delay and propagation vector in complex networks , 2012 .

[42]  Guihai Yu,et al.  More on the minimum skew-rank of graphs , 2015 .

[43]  Veerabahu Shanmugasundaram,et al.  Molecular similarity measures. , 2011, Methods in molecular biology.

[44]  Antal Járai,et al.  On the measurable solution of a functional equation arising in information theory , 1979 .

[45]  F Emmert-Streib,et al.  Networks for systems biology: conceptual connection of data and function. , 2011, IET systems biology.

[46]  Y. Moreno,et al.  Dynamic instability of cooperation due to diverse activity patterns in evolutionary social dilemmas , 2015, 1502.07724.

[47]  E. Ott,et al.  Onset of synchronization in large networks of coupled oscillators. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

[48]  Boris Furtula,et al.  On Zagreb Indices and Coindices , 2015 .

[49]  Robin J. Wilson,et al.  An Atlas of Graphs , 1999 .

[50]  Bohdan Zelinka,et al.  On a certain distance between isomorphism classes of graphs , 1975 .

[51]  Xueliang Li,et al.  A Survey on the Randic Index , 2008 .

[52]  Yongtang Shi,et al.  On a Conjecture about Tricyclic Graphs with Maximal Energy , 2013, 1312.0204.

[53]  Heping Zhang,et al.  Spectrum of Matching Forcing Numbers of a Hexagonal System with a Forcing Edge , 2015 .

[54]  D. Cvetkovic,et al.  Spectra of Graphs: Theory and Applications , 1997 .

[55]  David A. Gregory,et al.  Inertia and biclique decompositions of joins of graphs , 2003, J. Comb. Theory B.

[56]  Antonio Robles-Kelly,et al.  Edit distance from graph spectra , 2003, Proceedings Ninth IEEE International Conference on Computer Vision.

[57]  Kexiang Xu,et al.  A Survey on Graphs Extremal with Respect to Distance-Based Topological Indices , 2014 .

[58]  Hosam Abdo,et al.  Estimating the spectral radius of a graph by the second Zagreb index , 2014 .

[59]  Sezer Sorgun.  On Randic Energy of Graphs , 2014 .