The Critical Groups Of Graphs

The critical group of a connected graph is a finite abelian group whose structure is a subtle isomorphism, invariant of a graph. It is closely connected with the graph Laplacian. In this paper, we determined the structure of the critical group on the 3-circulant graphs and the products of cycle and some paths, we prove that(1) the critical group of Mobius ladder: if n = 2m + 1, thenwhere the sequence h_m is defined as h0 = 1, h₁ = 3, h_m = 4h_m1 — h_m2 for m ≥ 2 and if n = 2m and m is odd thenand if n = 2m is even and m is even thenwhere the sequence k_m is defined as k₀ = 1, k₁ = 2, k_m = 4k_m1 — k_m2 for m ≥2.(2) the critical group of P_n×C₃: K(P_n ×C₃) = Z_tn ⊕ Z_3(tn) where the sequence t_n is defined as t₀ = 0, t₁ = l,t_n = 5t_n-1 — t_n2.(3) the critical group of P₃ · C_n : K(P₃ · C_n) ≌Z₂²âŠ•Z₄^n-2 ⊕ Z₄n.(4) the critical group of P₄ ? C_n :if 3 | n, then K(P₄ · C_n) ≌Z_2(n,,hn) ⊕ Z_2(hn) ⊕ Z_2nhn/(n,hn);if 3 | n, then K(P₄ · C_n) ≌Z_2(n,hn)/3⊕ Z_2hn ⊕ Z_{(6nhn)/(n,hn)}.where the sequence h_n is defined as h₀ = 0, h₁ = 1, h_m = 4h_m1 - h_m2.(5) the critical group of Sm-Cn: K(Sm â–  Cn) = Z^²)n+2 ? Z^2 ? Z2mn.