On The Critical Group Of Bracelets Graph

The critical group of a connected graph is a finite abelian group whose structureis a subtle isomorphism invariant of the graph. It is closely connected with the graphLaplacian. In this paper, we determined the structure of the critical group on theCartesian products graph P_n×C₄ and bracelets graph C_n,3[(12)] and C_n,3[(123)], weproved that(1) the critical group of bracelets graph C_n,3[(12)] and C_n,3[(123)]:(a.1). For even integer n=2m, the critical group of C_n,3[(12)] is isomorphicto Z_n,sm⊕(Z_sm)²âŠ•Z_7sm⊕Z_21nsm/(n,sm) when 3(?)n and Z_(n,sm)/3⊕(Z_sm)²âŠ•Z_21sm⊕Z_21nsm/(n,sm)when 3|n.(a.2). For odd integer n=2m+1, the critical group of C_n,3[(12)] is isomorphicto Z_n,sn⊕Z_sn⊕Z_7nsn/(n,sn) when 3(?)n and Z_(n,sn)/3⊕Z_sn⊕Z_21nsN/(n,sn) when 3|n, wheres_n=5s_n-1-s_n-2, with initial value: s₀=0, s₁=1.(b.1). For n=3k and n=2m, the critical group of C_n,3[(123)] is isomorphicto Z_(n,sm)/3⊕(Z_sm)²âŠ•Z_21sm⊕Z_21nsm/(n,sm), where s_n=5s_n-1-s_n-2, with initial value:s₀=0, s₁=1.(b.2). For n=3k and n=2m+1, the critical group of C_n,3[(123)] is isomorphicto Z_(n.vm)/3⊕(Z_vm)²âŠ•Z_3vm⊕Z_3nvm/(n,vm), where v_n=5v_n-1-v_n-2, with initial value:v₀=1, v₁=6.(b.3). For n=3k+1 or n=3k+2, the critical group of C_n,3 [(123)] is isomorphicto Z_n,un⊕Z_un⊕Z_3nun/(n,un), where u_n=5u_n-1-u_n-2-1, with initial value u₀=1, u₁=2. (2)the critical group of P_n×C₄:when n≥2, we havewhere the sequence s_n is defined as s₀ =0, s₁=1, s_n=6s_n-1-s_n-2 for (n≥2);and sequence t_n is defined as t₀=0, t₁=1, t_n=4t_n-1-t_n-2 for (n≥2).