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cmsy101vrerticesisn2n 2̹.!WVeendowthesetTn coflabSeledtreesonn41vrerticeswithuniformprobabilityV,givingweightn22 nottoeachtree. Eacrh0treeinTncomeswithitsincidencematrix,Uthen.n0symmetricmatrixAwithenrtryij-equalto1ifthereisanedgebSetweenverticesiandWj_andto0otherwise.Eacrhsuchmatrixhasn(real)eigenvXalues,whicrhcbydenitionformthespSectrumofthecorrespondingtree.+Thisleadsinturntonn2n 2T=n2n 1M˹eigenrvXaluescountedwithmultiplicityforMTnDasawhole.Inthesequel,mLwrewillconcentrateonthemultiplicityof^theeigenrvXalue0.LetZ ܞ(Tƹ)bSethemultiplicityoftheeigenvXalue0intheBspSectrumoftheincidencematrixofthetreeTƹ,i.e.dthedimensionof\thekrernel. FVoreachnUR1,ythe\restrictionZn ofZ99toTnisarandom
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