GraphData["Classes"]

GraphData[{"WhiteBishop", {3, 5}}]
CG=GraphData[{"WhiteBishop", {3, 5}},"ChromaticPolynomial"];
CG[x]//Expand

GraphData[{"CocktailParty",4}]
CG=GraphData[{"CocktailParty",4},"ChromaticPolynomial"];
CG[x]//Expand

GraphData[{"BlanusaSnark",{1,1}}]

Cases[GraphData["Archimedean"],_String]

CG=GraphData[{"Cage",{4,6}},"ChromaticPolynomial"];
CG[x]//Factor
CG[x]//Expand

GraphData["Archimedean","AllImages"]

GraphData["Kuratowski","AllImages"]

ClearAll[CG];
CG=GraphData[{"Lollipop", {3, 3}},"ChromaticPolynomial"]

GraphData[{"Lollipop", {3, 3}}]

CG[x]//Factor//Expand//TraditionalForm

ChromaticPolynomial[{1\[UndirectedEdge]2},x]

ChromaticPolynomial[CompleteGraph[3],x]//Factor//TraditionalForm

ClearAll[g];
g=Graph[{1\[UndirectedEdge]2,2\[UndirectedEdge]3,3\[UndirectedEdge]4,4\[UndirectedEdge]2,1\[UndirectedEdge]3,5\[UndirectedEdge]4,1\[UndirectedEdge]4},VertexLabels->"Name"]

ChromaticPolynomial[g,x]

g=Graph[{1\[UndirectedEdge]2,2\[UndirectedEdge]3},"ChromaticPolynomial"]

ClearAll[g];
g=Graph[{1\[UndirectedEdge]2,2\[UndirectedEdge]3,3\[UndirectedEdge]5,1\[UndirectedEdge]5,1\[UndirectedEdge]3},VertexLabels->"Name"]
ChromaticPolynomial[g,x]//TraditionalForm

ClearAll[g];
g=Graph[{1\[UndirectedEdge]2,2\[UndirectedEdge]3,2\[UndirectedEdge]4,2\[UndirectedEdge]5,1\[UndirectedEdge]4},VertexLabels->"Name"]
ChromaticPolynomial[g,x]//Factor//TraditionalForm

ClearAll[g];
g=Graph[{1\[UndirectedEdge]2,2\[UndirectedEdge]3,3\[UndirectedEdge]4,1\[UndirectedEdge]4,2\[UndirectedEdge]5,3\[UndirectedEdge]5,4\[UndirectedEdge]5},VertexLabels->"Name"]
ChromaticPolynomial[g,x]//Factor

ClearAll[g];
g=Graph[{1\[UndirectedEdge]2,2\[UndirectedEdge]3,3\[UndirectedEdge]4,1\[UndirectedEdge]4,2\[UndirectedEdge]5,3\[UndirectedEdge]5,4\[UndirectedEdge]5,1\[UndirectedEdge]6,6\[UndirectedEdge]7,7\[UndirectedEdge]8,8\[UndirectedEdge]9,9\[UndirectedEdge]6},VertexLabels->"Name"]
ChromaticPolynomial[g,x]//Factor

x(x-1)(((x-2))^2)(((x-1))^3)-x(x-1)(((x-2))^3)(x-1)//Factor

x(((x-1))^5)(x-2)((x^2)-5x+7)-x(((x-1))^3)(((x-2))^2)((x^2)-5x+7)//Factor

(1/x)(x(((x-1))^2)(((x-2))^2)+x(((x-1))^3))*(x(x-1)(x-2)((x^2)-3x+3)-x(x-1)(((x-2))^2))//Simplify

%//Factor

Needs["Combinatorica`"];

ClearAll[g];
g=WheelGraph[5]
ChromaticPolynomial[g,x]//Factor

x(x-2)^4 + x^2 - 2x //Expand//Factor

x(x-1)(x-2)^2(x^2-3x+3)-%9

%19//Factor

ClearAll[g];
g=Graph[{1\[UndirectedEdge]2,2\[UndirectedEdge]3,3\[UndirectedEdge]4,1\[UndirectedEdge]4,2\[UndirectedEdge]6,3\[UndirectedEdge]6,4\[UndirectedEdge]5,1\[UndirectedEdge]5,5\[UndirectedEdge]6},VertexLabels->"Name"]
ChromaticPolynomial[g,x]//Factor

%18//Expand

Falling Factorial

4*FactorialPower[x,3]+3*FactorialPower[x,4]-3*(x(((x-1))^3)(x-2)-x(((x-1))^2)(x-2))//FunctionExpand//Expand

(x-1)(x(((x-2))^3)-x(x-2)(x-3)-x)//Expand

%21-((((-2+x))^2) (-1+x) x (3-3 x+(x^2)))+

ClearAll[g];
g=Graph[{1\[UndirectedEdge]2,2\[UndirectedEdge]3,3\[UndirectedEdge]4,4\[UndirectedEdge]5,5\[UndirectedEdge]6,6\[UndirectedEdge]1,2\[UndirectedEdge]5,3\[UndirectedEdge]5,3\[UndirectedEdge]6,1\[UndirectedEdge]4},VertexLabels->"Name"]
ChromaticPolynomial[g,x]//Factor

ChromaticPolynomial[WheelGraph[5],x]//Factor

x(x-1)(((x-2))^2)((x^2)-4x+6)-%108

%//Factor

ClearAll[gr,a,b,c,d,e,f,g,h,i];
gr=Graph[{a\[UndirectedEdge]b,b\[UndirectedEdge]c,c\[UndirectedEdge]d,d\[UndirectedEdge]e,e\[UndirectedEdge]f,f\[UndirectedEdge]a,d\[UndirectedEdge]b,d\[UndirectedEdge]a,c\[UndirectedEdge]g,g\[UndirectedEdge]h,h\[UndirectedEdge]d,d\[UndirectedEdge]g,b\[UndirectedEdge]e,f\[UndirectedEdge]c,c\[UndirectedEdge]i},VertexLabels->"Name"
,VertexCoordinates->{{0,1},{1,1},{2,1},{2,0},{1,0},{0,0},{3,1},{3,0},{1,2}}]
ChromaticPolynomial[gr,x]//Factor

ClearAll[gr,a,b,c,d,e,f,g,h,i];
gr=Graph[{a\[UndirectedEdge]b,b\[UndirectedEdge]c,c\[UndirectedEdge]d,d\[UndirectedEdge]e,e\[UndirectedEdge]f,f\[UndirectedEdge]a,d\[UndirectedEdge]b,d\[UndirectedEdge]a,b\[UndirectedEdge]e,f\[UndirectedEdge]c},VertexLabels->"Name"
,VertexCoordinates->{{0,1},{1,1},{2,1},{2,0},{1,0},{0,0}}]
ChromaticPolynomial[gr,x]//Factor

ClearAll[gr,a,b,c,d,e,f,g,h,i];
gr=Graph[{a\[UndirectedEdge]b,b\[UndirectedEdge]c,c\[UndirectedEdge]d,d\[UndirectedEdge]e,e\[UndirectedEdge]f,f\[UndirectedEdge]a,d\[UndirectedEdge]b,d\[UndirectedEdge]a,c\[UndirectedEdge]g,g\[UndirectedEdge]h,h\[UndirectedEdge]d,d\[UndirectedEdge]g,b\[UndirectedEdge]e,f\[UndirectedEdge]c,c\[UndirectedEdge]a},VertexLabels->"Name"
,VertexCoordinates->{{0,1},{1,0.75},{2,1},{2,0},{1,0},{0,0},{3,1},{3,0}}]
ChromaticPolynomial[gr,x]//Factor

%232-%220

%//Factor

%/.x->4

ClearAll[gr,a,b,c,d,e,f,g,h,i];
gr=Graph[{a\[UndirectedEdge]b,b\[UndirectedEdge]c,c\[UndirectedEdge]d,d\[UndirectedEdge]e,e\[UndirectedEdge]f,f\[UndirectedEdge]a,d\[UndirectedEdge]b,d\[UndirectedEdge]a,c\[UndirectedEdge]g,g\[UndirectedEdge]h,h\[UndirectedEdge]d,d\[UndirectedEdge]g,b\[UndirectedEdge]e,f\[UndirectedEdge]c},VertexLabels->"Name"
,VertexCoordinates->{{0,1},{1,1},{2,1},{2,0},{1,0},{0,0},{3,1},{3,0},{1,2}}]
ChromaticPolynomial[gr,x]//Factor

ClearAll[gr,a,b,c,d,e,f,g,h,i];gr=Graph[{a\[UndirectedEdge]b,b\[UndirectedEdge]c,c\[UndirectedEdge]d,d\[UndirectedEdge]e,e\[UndirectedEdge]f,f\[UndirectedEdge]a,d\[UndirectedEdge]b,d\[UndirectedEdge]a,c\[UndirectedEdge]g,g\[UndirectedEdge]h,h\[UndirectedEdge]d,d\[UndirectedEdge]g,b\[UndirectedEdge]e,f\[UndirectedEdge]c},VertexLabels->"Name",VertexCoordinates->{{0,1},{1,1},{2,1},{2,0},{1,0},{0,0},{3,1},{3,0}}]
ChromaticPolynomial[gr,x]//Factor

ClearAll[gr,a,b,c,d,e,f,g,h,i];gr=Graph[{a\[UndirectedEdge]b,b\[UndirectedEdge]c,c\[UndirectedEdge]d,d\[UndirectedEdge]e,e\[UndirectedEdge]f,f\[UndirectedEdge]a,d\[UndirectedEdge]b,d\[UndirectedEdge]a,c\[UndirectedEdge]g,g\[UndirectedEdge]h,h\[UndirectedEdge]d,d\[UndirectedEdge]g,b\[UndirectedEdge]e,f\[UndirectedEdge]c,a\[UndirectedEdge]i,c\[UndirectedEdge]i},VertexLabels->"Name",VertexCoordinates->{{0,1},{1,1},{2,1},{2,0},{1,0},{0,0},{3,1},{3,0},{1,1.5}}]
ChromaticPolynomial[gr,x]//Factor

ClearAll[gr,a,b,c,d,e,f,g,h,i];gr=Graph[{a\[UndirectedEdge]b,b\[UndirectedEdge]c,c\[UndirectedEdge]d,d\[UndirectedEdge]e,e\[UndirectedEdge]f,f\[UndirectedEdge]a,d\[UndirectedEdge]b,d\[UndirectedEdge]a,c\[UndirectedEdge]g,g\[UndirectedEdge]h,h\[UndirectedEdge]d,d\[UndirectedEdge]g,b\[UndirectedEdge]e,f\[UndirectedEdge]c,c\[UndirectedEdge]i},VertexLabels->"Name",VertexCoordinates->{{0,1},{1,1},{2,1},{2,0},{1,0},{0,0},{3,1},{3,0},{1,1.5}}]
ChromaticPolynomial[gr,x]//Factor

ClearAll[gr,a,b,c,d,e,f,g,h,i];
gr=Graph[{a\[UndirectedEdge]b,b\[UndirectedEdge]c,c\[UndirectedEdge]d,d\[UndirectedEdge]e,e\[UndirectedEdge]f,f\[UndirectedEdge]a,d\[UndirectedEdge]b,d\[UndirectedEdge]a,c\[UndirectedEdge]g,g\[UndirectedEdge]h,h\[UndirectedEdge]d,d\[UndirectedEdge]g,b\[UndirectedEdge]e,f\[UndirectedEdge]c,c\[UndirectedEdge]a},VertexLabels->"Name"
,VertexCoordinates->{{0,1},{1,0.75},{2,1},{2,0},{1,0},{0,0},{3,1},{3,0}}]
ChromaticPolynomial[gr,x]//Factor

ClearAll[gr,a,b,c,d,e,f,g,h,i];gr=Graph[{a\[UndirectedEdge]b,b\[UndirectedEdge]c,c\[UndirectedEdge]d,d\[UndirectedEdge]e,e\[UndirectedEdge]f,f\[UndirectedEdge]a,d\[UndirectedEdge]b,d\[UndirectedEdge]a,c\[UndirectedEdge]g,g\[UndirectedEdge]h,h\[UndirectedEdge]d,d\[UndirectedEdge]g,b\[UndirectedEdge]e,f\[UndirectedEdge]c,a\[UndirectedEdge]i,c\[UndirectedEdge]i},VertexLabels->"Name",VertexCoordinates->{{0,1},{1,1},{2,1},{2,0},{1,0},{0,0},{3,1},{3,0},{1,1.5}}]
ChromaticPolynomial[gr,x]//Factor

EdgeAdd[EdgeDelete[%34,{a\[UndirectedEdge]b,b\[UndirectedEdge]c,b\[UndirectedEdge]d,b\[UndirectedEdge]e}],{a\[UndirectedEdge]e,c\[UndirectedEdge]e}]

ChromaticPolynomial[%42,x]//Factor