Potential Analysis

, Volume 25, Issue 3, pp 259–268 | Cite as

Non-bipartiteness of Graphs and the Upper Bounds of Dirichlet Forms



The sum of the lower bound and the upper one of the spectrum of our discrete Laplacian is less than or equal to 2. The equality holds if a graph is bipartite while the converse does not hold for general infinite graphs. In this paper, we give an estimate of the upper bounds of Dirichlet forms and using this estimate together with an h-transform, we show that the sum is strictly less than 2 for a certain class of infinite graphs.

Mathematics Subject Classifications (2000)

Primary 47B39, 58J50 Secondary 05C75 

Key words

discrete Laplacian Dirichlet form bipartiteness h-transform 


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Copyright information

© Springer Science + Business Media B.V. 2006

Authors and Affiliations

  1. 1.Mathematics Laboratories, College of Arts and SciencesShowa UniversityFujiyoshidaJapan
  2. 2.Faculty of MathematicsKyushu UniversityHigashi-kuJapan

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