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On the Critical Group of Bracelets Graph

Author: ShenJin
Tutor: HouYaoPing
School: Hunan Normal University
Course: Operational Research and Cybernetics
Keywords: The Laplacian matrix Critical Groups Sandpile group Smith normal form of the matrix Bracelets Fig.
CLC: O157.5
Type: Master's thesis
Year: 2007
Downloads: 15
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Abstract


The connected graph critical group is a number of Spanning refinable, it is defined in the diagram on a finite abelian group. Its group structure is a fine invariant, it is closely related to the Laplacian theory. n × C 4 critical group, the following conclusions: (1) bracelet graph is defined as follows: given a constant p ∈ Z, p ≥ 2. Given integer n ≥ 3 and n permutation σ 1 σ 2 , ..., and σ . n a complete graph K b , they might be to 1,2, ..., n label. Edges between them in accordance with the following rules: i-K b vertex v in the first (i 1) K b vertex σ i (v) with edge. N 1 = 1 can be easily considered. So we get a bracelet Figure denoted G n, b 1 σ 2 ... the σ n ]. The structure of the critical group: (a.1). When n = 2m and 3 (?) N the G n 3 [(12)] the critical group the the Z- n, s , m ⊕ (Z s m ) 2 ⊕ Z 7s m ⊕ Z < sub> (21ns m ) / (n, s m ) . When n = 2m and | n, G (n, 3) [(12)] the critical group Z the (n, s , m ) / 3 ⊕ ( Z s m ) 2 ⊕ Z 21s m ⊕ Z ( 21ns m ) / (n, s m ) . Here the s n = 5s n-1 -s n-2 , the initial: s by 0 = 0 , s 1 = 1. (A.2). When n = 2m 1 and 3 (?) N the G n 3 [(12)] the critical group is Z (n, s , N ) ⊕ Z < sub> s n ⊕ Z 7ns n / (n, s n ) . When n = 2m 1 and 3 | N, G N, 3 [(12)] the critical group for the Z the (n, s , N ) / 3 ⊕ Z s n ⊕ Z 21ns n / (n, s n ) . Here the s n = 5s n-1 -s n-2 , the initial: s by 0 = 0 , s 1 = 1. (B.1). When n = 3k and N = 2m, G n, 3 [(123])] the critical group for the Z the (n, s , m ) / 3 ⊕ (Z s m ) 2 ⊕ Z 21s m ⊕ Z 21ns m / (n, s m ) . Here the s n = 5s n-1 -s n-2 , the initial: s by 0 = 0 , s 1 = 1. (B.2). When n = 3k and n = 2M 1 the G n 3 [(123)] the critical group as Z (n, v m ) / 3 ⊕ (Z v m ) 2 ⊕ Z 3v m ⊕ Z 3nv m / (n, v m ) . The v n = 5v n-1 -v n-2 , the initial value as follows: v 0 = 1, v 1 = 6. (B.3). When n = 3k 1 or n = 3k 2, the the G n, 3 [(123)] the critical group: Z n, u n ⊕ Z u n ⊕ Z 3nu n / (u, u n ) . Here u n = 5u n-1 -u n-2 -1, initial u 0 = 1, u 1 = 2. (2) P n × C 4 critical group structure: when n ≥ 2, defined as the sequence s n s 0 = 0, s 1 = 1, s n = 6s n-1 -s n -2 (n ≥ 2); sequence t n is defined as T 0 = 0, t 1 = 1, t n = 4t n-1 -t n-2 (n ≥ 2).

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CLC: > Mathematical sciences and chemical > Mathematics > Algebra,number theory, portfolio theory > Combinatorics ( combinatorics ) > Graph Theory
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