Principal eigenvectors of irregular graphs

By Sebastian M. Cioaba, David A. Gregory. 2007

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Let G be a connected graph. Thispaper s extreme entriesof the principal eigenvector x of G, the unique positive unit eigenvector corresponding to the greatest eigenvalue λ1 of the adjacency matrix of G.I fG hasmaximum degree ∆, the greates t entry xmax of x isat mos t 1/ 1+ λ 2 /∆. This improves a result of Papendieck and Recht. The least entry xmin of x aswell asthe principal ratio xmax/xmin are studied. It is conjectured that for connected graphs of order n ≥ 3, the principal ratio isalwaysattained by one of the lollipop graphsobtained by attaching a path graph to a vertex of a complete graph.

Subfields: Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/80f3093b3fd2a95b35d9af994d22a548a0ef9475