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Characteristic polynomials of ramified uniform covering digraphs
 European Journal of Combinatorics
"... We give a decomposition formula for the characteristic polynomials of ramified uniform covers of digraphs. Similarly, we obtain a decomposition formula for the characteristic polynomials of ramified regular covers of digraphs. As applications, we establish decomposition formulas for the characterist ..."
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We give a decomposition formula for the characteristic polynomials of ramified uniform covers of digraphs. Similarly, we obtain a decomposition formula for the characteristic polynomials of ramified regular covers of digraphs. As applications, we establish decomposition formulas for the characteristic polynomials of branched covers of digraphs and the zeta functions of ramified covers of digraphs. Key words: characteristic polynomial, adjacency matrix, voltage digraph, ramified uniform cover, ramified regular cover, zeta function 1
Bartholdi zeta functions of group coverings of digraphs
 Far East J. Math. Sci
, 2005
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A New Bartholdi Zeta Function of a Graph
"... We define a new type of the Bartholdi zeta function of a graph G, and give a determinant expression of it. Furthermore, we define a new type of the Bartholdi Lfunction of G, and present a determinant expression for a new type of the Bartholdi Lfunction of G. As a corollary, we show that a new type ..."
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We define a new type of the Bartholdi zeta function of a graph G, and give a determinant expression of it. Furthermore, we define a new type of the Bartholdi Lfunction of G, and present a determinant expression for a new type of the Bartholdi Lfunction of G. As a corollary, we show that a new type of the Bartholdi zeta function of a regular covering of G is a product of new Batholdi Lfunctions of G.
Bartholdi Zeta Functions of Fractal Graphs
"... Recently, Guido, Isola and Lapidus [11] defined the Ihara zeta function of a fractal graph, and gave a determinant expression of it. We define the Bartholdi zeta function of a fractal graph, and present its determinant expression. 1 ..."
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Recently, Guido, Isola and Lapidus [11] defined the Ihara zeta function of a fractal graph, and gave a determinant expression of it. We define the Bartholdi zeta function of a fractal graph, and present its determinant expression. 1
Eigenvalues of Nonbacktracking Walks in a Cycle with Random Loops
, 2007
"... In this paper we take a very special model of a random nonregular graph and study its nonbacktracking spectrum. We study graphs consisting of a cycle with some random loops added; the graphs are not regular and their nonbacktracking spectrum does not seem to be confined to some onedimensional se ..."
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In this paper we take a very special model of a random nonregular graph and study its nonbacktracking spectrum. We study graphs consisting of a cycle with some random loops added; the graphs are not regular and their nonbacktracking spectrum does not seem to be confined to some onedimensional set in the complex plane. The nonbacktracking spectrum is required in some applications, and has no straightforward connection to the usual adjacency matrix spectrum for general graphs, unlike the situation for regular graphs. Experimentally, the random graphs ’ spectrum appears similar in shape to its deterministic counterpart, but differs because the eigenvalues are visibly clustered, especially with a mysterious gap around Re(λ) = 1.
The Scattering Matrix of a Graph
"... Recently, Smilansky expressed the determinant of the bond scattering matrix of a graph by means of the determinant of its Laplacian. We present another proof for this Smilansky’s formula by using some weighted zeta function of a graph. Furthermore, we reprove a weighted version of Smilansky’s formul ..."
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Recently, Smilansky expressed the determinant of the bond scattering matrix of a graph by means of the determinant of its Laplacian. We present another proof for this Smilansky’s formula by using some weighted zeta function of a graph. Furthermore, we reprove a weighted version of Smilansky’s formula by Bass ’ method used in the determinant expression for the Ihara zeta function of a graph. 1
Zeta Functions and Chaos
, 2009
"... Abstract: The zeta functions of Riemann, Selberg and Ruelle are briefly introduced along with some others. The Ihara zeta function of a finite graph is our main topic. We consider two determinant formulas for the Ihara zeta, the Riemann hypothesis, and connections with random matrix theory and quant ..."
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Abstract: The zeta functions of Riemann, Selberg and Ruelle are briefly introduced along with some others. The Ihara zeta function of a finite graph is our main topic. We consider two determinant formulas for the Ihara zeta, the Riemann hypothesis, and connections with random matrix theory and quantum chaos. 1
Documenta Math. 1243 The Zeta Function of a Finite Category
, 2012
"... Abstract. We define the zetafunction ofafinite category. We prove a theorem that states a relationship between the zeta function of a finite category and the Euler characteristic of finite categories, called the series Euler characteristic [BL08]. Moreover, it is shown that for a covering of finite ..."
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Abstract. We define the zetafunction ofafinite category. We prove a theorem that states a relationship between the zeta function of a finite category and the Euler characteristic of finite categories, called the series Euler characteristic [BL08]. Moreover, it is shown that for a covering of finite categories, P: E → B, the zeta function of E is that of B to the power of the number of sheets in the covering. This is a categorical analogue of the unproved conjecture of Dedekind for
CUTOFF ON ALL RAMANUJAN GRAPHS
"... Abstract. We show that on every Ramanujan graph G, the simple random walk exhibits cutoff: when G has n vertices and degree d, the totalvariation distance of the walk from the uniform distribution at time t = d d−2 logd−1 n + s logn is asymptotically P(Z> c s) where Z is a standard normal variab ..."
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Abstract. We show that on every Ramanujan graph G, the simple random walk exhibits cutoff: when G has n vertices and degree d, the totalvariation distance of the walk from the uniform distribution at time t = d d−2 logd−1 n + s logn is asymptotically P(Z> c s) where Z is a standard normal variable and c = c(d) is an explicit constant. Furthermore, for all 1 ≤ p ≤ ∞, dregular Ramanujan graphs minimize the asymptotic Lpmixing time for SRW among all dregular graphs. Our proof also shows that, for every vertex x in G as above, its distance from n − o(n) of the vertices is asymptotically logd−1 n. 1.
MARKOV PATHS, LOOPS AND FIELDS (Preliminary version)
, 2008
"... We study the Poissonnian ensembles of Markov loops and the associated renormalized self intersection local times. 1 ..."
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We study the Poissonnian ensembles of Markov loops and the associated renormalized self intersection local times. 1