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  "body": "\n## Definition\n\nLet $M$ be a [[matroid|PX31ZNYS3Z]] on a finite ground set $E$ with [[rank function|SYRCTH853H]] $r$.\nThe *Tutte polynomial* of $M$ is\n$$T_M(x, y) = \\sum_{A \\subseteq E} (x - 1)^{r(E) - r(A)} \\, (y - 1)^{\\abs{A} - r(A)}.$$\nLet $G$ be a finite [[graph|00C0BR7D7H]], with loops and multiple edges allowed.\nIts Tutte polynomial $T_G$ is the Tutte polynomial of its graphic matroid:\n$$T_G(x, y) = \\sum_{A \\subseteq E(G)} (x - 1)^{k(A) - k(E)} \\, (y - 1)^{k(A) + \\abs{A} - \\abs{V(G)}},$$\nwhere $k(A)$ is the number of connected components of the graph $(V(G), A)$.\n\n## Remarks\n\nLet $e \\in E$, and let $M \\setminus e$ and $M / e$ be the [[deletion and contraction|4WCYBSH1P6]] of $e$.\nIf $e$ is neither a loop nor a coloop, then $T_M = T_{M \\setminus e} + T_{M / e}$.\nIf $e$ is a coloop, then $T_M = x \\, T_{M / e}$; if $e$ is a loop, then $T_M = y \\, T_{M \\setminus e}$.\nFor the dual matroid, $T_{M^*}(x, y) = T_M(y, x)$.\nThe value $T_M(1, 1)$ is the number of bases of $M$.\nFor a graph $G$, the [[chromatic polynomial|0B621RBVS9]] is $P_G(\\lambda) = (-1)^{\\abs{V(G)} - k(E)} \\lambda^{k(E)} \\, T_G(1 - \\lambda, 0)$.\n\nExample: for the triangle $K_3$, $\\abs{V} = 3$ and $k(E) = 1$.\nThe empty set gives $(x-1)^2$, the $3$ single edges give $3(x-1)$, the $3$ pairs give $3$, and the full edge set gives $y - 1$.\nThe sum is $x^2 - 2x + 1 + 3x - 3 + 3 + y - 1 = x^2 + x + y$.\nThen $T(1, 1) = 3$ spanning trees, and $(-1)^2 \\lambda \\, T(1 - \\lambda, 0) = \\lambda(\\lambda - 1)(\\lambda - 2)$ is the chromatic polynomial of $K_3$.\n\nNon-example: the rank generating polynomial $\\sum_{A} u^{r(E) - r(A)} v^{\\abs{A} - r(A)}$ is not the Tutte polynomial itself; the Tutte polynomial is obtained from it by the substitution $u = x - 1$, $v = y - 1$.\n",
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