Non-bipartiteness of Graphs and the Upper Bounds of Dirichlet Forms |
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Authors: | Yusuke Higuchi Tomoyuki Shirai |
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Institution: | (1) Mathematics Laboratories, College of Arts and Sciences, Showa University, 4562 Kamiyoshida, Fujiyoshida Yamanashi, 403-0005, Japan;(2) Faculty of Mathematics, Kyushu University, Hakozaki 6-10-1, Higashi-ku Fukuoka, 812-8581, Japan |
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Abstract: | 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.Dedicated to Professor Yoichiro Takahashi on his sixtieth birthday. |
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Keywords: | Primary 47B39 58J50 Secondary 05C75 |
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