| Budget Amount *help |
¥2,600,000 (Direct Cost: ¥2,000,000、Indirect Cost: ¥600,000)
Fiscal Year 2010: ¥520,000 (Direct Cost: ¥400,000、Indirect Cost: ¥120,000)
Fiscal Year 2009: ¥520,000 (Direct Cost: ¥400,000、Indirect Cost: ¥120,000)
Fiscal Year 2008: ¥520,000 (Direct Cost: ¥400,000、Indirect Cost: ¥120,000)
Fiscal Year 2007: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
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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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