Extremal Properties of Graphs and Eigencentrality in Trees with a Given Degree Sequence

REFERENCES

• Alderson , D. L. & Li , L. ( 2007 ). Diversity of graphs with highly variable connectivity . Physical Review E , 75 ( 046102 ), 1 – 11 .
   Google Scholar
• Berge , C. ( 1985 ). Graphs () , 3rd ed. , Amsterdam : North-Holland .
   Google Scholar
• Biyikoglu , T. & Leydold , J. ( 2006 ). Largest Eigenvalues of Degree Sequences . (Research Report Series No. 35) . Wien , Austria : Department of Statistics and Mathematics of Wien .
   Google Scholar
• Bollobas , B. & Erdos , P. ( 1998 ). Graphs with extremal weights . Ars Combinatoria , 50 , 225 – 233 .
   Web of Science ®Google Scholar
• Bonacich , P. ( 1972 ). A technique for analyzing overlapping memberships . In H. Costner , (Ed.), Sociological Methodology , (pp. 176 – 185 ). San Francisco : Jossey-Bass .
   Google Scholar
• Bonacich , P. ( 2007 ). Some unique properties of eigenvector centrality . Social Networks , 29 , 555 – 564 .
   Web of Science ®Google Scholar
• Delorme , C. , Favaron , O. , & Rautenbach , D. ( 2003 ). Closed formulas for the numbers of small independent sets and matchings and an extremal problem for trees . Discrete Applied Mathematics , 130 , 503 – 512 .
   Google Scholar
• Erdos , P. & Gallai , T. ( 1960 ). Graphs with prescribed degrees of vertices [Hungarian] . Matematikai Lapok , 11 , 264 – 274 .
   Google Scholar
• Everett , M. G. & Borgatti , S. P. ( 2000 ). Peripheries of cohesive subsets . Social Networks , 21 , 397 – 407 .
   Google Scholar
• Grassi , R. , Stefani , S. , & Torriero , A. ( 2007 ). Some new results on the eigenvector centrality . Journal of Mathematical Sociology , 31 , 237 – 248 .
   Web of Science ®Google Scholar
• Gutman , I. & Das , K. C. ( 2004 ). The first Zagreb index 30 years after . MATCH Communications in Mathematical and in Computer Chemistry , 583 – 592 .
   Google Scholar
• Gutman , I. & Furtula , B. ( 2008 ). Recent Results in the Theory of Randic Index . (Mathematical Chemistry Monograph No. 6) . Kragujevac , Serbia : University of Kragujevac .
   Google Scholar
• Harary , F. ( 1969 ). Graph Theory . Reading , MA : Addison-Wesley .
   Google Scholar
• Hardy , G. H. , Littlewood , J. E. , & Pólya , G. ( 1988 ). Inequalities (2nd ed.) . Cambridge , UK : Cambridge University Press .
   Google Scholar
• Horn , R. A. & Johnson , C. R. ( 1985 ). Matrix Analysis . Cambridge , UK : Cambridge University Press .
   Google Scholar
• Li , L. , Alderson , D. , Doyle , J. C. , & Willinger , W. ( 2005a ). Supplemental material: the S(G) metric and assortativity . Internet Mathematics , 2 ( 4 ), 1 – 6 .
   Google Scholar
• Li , L. , Alderson , D. , Doyle , J. C. , & Willinger , W. ( 2005b ). Toward a theory of scale-free graphs: definitions, properties and implications . Internet Mathematics , 2 ( 4 ), 431 – 523 .
   Google Scholar
• Lovász , L. ( 1993 ). Combinatorial Problems and Exercises () , 2nd ed. . Amsterdam : North-Holland .
   Google Scholar
• Lu , M. , Liu , H. , & Tian , F. ( 2004 ). The connectivity index . MATCH Communications in Mathematical and Computer Chemistry , 51 , 149 – 154 .
   Google Scholar
• Marshall , A. W. & Olkin , I. ( 1979 ). Inequalities: Theory of Majorization and Its Applications . New York : Academic Press .
   Google Scholar
• Newman , M. E. J. ( 2002 ). Assortative mixing in networks . Physical Review Letters , 89 ( 20 ), 1 – 4 .
   Web of Science ®Google Scholar
• Nikolić , S. , Kovačević , G. , Miličević , A. , & Trinajstić , N. ( 2003 ). The Zagreb indices 30 years after . Croatica Chemica Acta , 76 , 113 – 124 .
   Web of Science ®Google Scholar
• Wang , H. ( 2008 ). Extremal trees with given degree sequence for the Randic index . Discrete Mathematics , 308 , 3407 – 3411 .
   Google Scholar