2010 Fiscal Year Final Research Report
Zeta function of a graph and its generalization
| Project/Area Number |
19540154
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
General mathematics (including Probability theory/Statistical mathematics)
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| Research Institution | Oyama National College of Technology |
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Principal Investigator |
SATOU Iwao Oyama National College of Technology, 一般科, 教授 (70154036)
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| Project Period (FY) |
2008 – 2010
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| Keywords | 応用数学 / 離散数学 |
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| Research Abstract |
At first, we introduced the (first)weighted Bartholdi zeta function of a digraph as a generalization of various zeta functions of a graph or a digraph, and presented its determinant expression. As corollaries, we obtained determinant expressions for various zeta functions of a graph or a digraph. Furthermore, we treated the second weighted (Bartholdi) zeta functions of a graph or a digraph.
Next, we extended the notion of a regular covering of a graph to a digraph, and gave decomposition formulas for the characteristic polynomial, the zeta function and the Bartholdi zeta function of a ramified covering of a digraph. Furthermore, we reformulated the Galois theory of graph coverings by Stark and Terras by voltage assignments. As an application, we presented a decomposition formula for the weighted Bartholdi zeta function of a quotient of a regular covering of a graph, and generalized it to a digraph.
We extended the results on the Ihara zeta functions of infinite graphs(periodic simple graph, periodic graph, fractal graph). Furthermore, we defined the Bartholdi zeta function for a hypergraph, and gave its determinant expression.
By using a determinant expression of the second weighted zeta function of a graph, we obtained a new proof the result on the scattering matrix of a graph by Smilansky.
Finally, we obtained the results on the weighted complexities of a digraph by using the weighted zeta function of a digraph.
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Research Products
(53 results)
