(* Content-type: application/mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 7.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 145, 7] NotebookDataLength[ 105575, 2863] NotebookOptionsPosition[ 103042, 2778] NotebookOutlinePosition[ 103534, 2797] CellTagsIndexPosition[ 103491, 2794] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"2", "-", "2"}]], "Input", CellChangeTimes->{{3.526484418443427*^9, 3.526484418660046*^9}}], Cell[BoxData["0"], "Output", CellChangeTimes->{3.526484427761094*^9, 3.526493425248184*^9, 3.526865840345376*^9, 3.526865974443255*^9, 3.526867032114827*^9, 3.52824558987403*^9}] }, Open ]], Cell[CellGroupData[{ Cell["Perfectly Alternating Graphs", "Title", CellChangeTimes->{{3.526484510834814*^9, 3.526484519642829*^9}}], Cell["\<\ Nicholas Wheeler 1 October 2011\ \>", "Text", CellChangeTimes->{{3.526484527040793*^9, 3.526484575915209*^9}}], Cell[CellGroupData[{ Cell["Introduction", "Subsection", CellChangeTimes->{{3.5264846058715982`*^9, 3.5264846078707533`*^9}}], Cell["\<\ I construct here a few examples of graphs that acquire interest from this \ fact: random walkers on such graphs always know\[LongDash]from the color of \ the node on which they stand\[LongDash]whether they have executed an even or \ odd number of steps in arriving at that node. 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