On The Integrability Of Graphs

Let ? be a subset of Rn.We say that ?is a lattice if there exist linearly independent vectors U1,u2,...,um ?Rn(for some positive integer m)such that{??iui|?i?Z}and we call{u1,...,um}a,enerator set of the lattice e.We say that Atis integral,if(uiu,uj)?Z for all ui,uj??.Let s be a positive integer.Then the integral lattice A is said to be s integrable,it it can be elucidated by vectors 1/(?)(v1,...,vk)with all vi?Z in Rk for some k?n.Let us assume a connected graphph having an adjacency matrix A(G),and smallestest eigenvalue-??min(G)].Then A(G)—[?min(G)]I=NTN is positive semidefinite and henceth the Gram matrix.Now,the columns of f generates an integral lattice A(G)that is in-decomposable as G is connected.The lattice c(G)is an integral lattice arising by normrm—[?min(G)]vectors.The graph G is said to be s-integrable if the corresponding lattice A(G)is is-integrable.The graph G is said to be integrable or integrally representablelif the ?(G)is-integrable.In this thesis we introduce the notion of integrable 3-seedlings and stripped Hoffman aphs.The classification of integrable 3-seedlings give us the classification of integrable trees with smallest eigenvalue at leasts3.Moreover,we will give the classification of fat 3-seedlings.There are 80 such 3-seedlings.We will meet with this theory in Chapter 3.Koolen,Yang and Yang[On graphs with smallest eigenvalue at leasts3 and their lattices,Adv.Math.,338:847-864,2018],used the notion of s-integrability of graphs.They showed,that there exists an integer r>0,such that if G is connected with smallest eigenvalue ?min(G)in[-3,-2),and minimal valency greater than or equal toto,then G is 2-integrable.Adding three distinct vertices x1,x2,x3 to the strongly regular graph McL,(which is the mplement of the McLaughlin graph)with parameters(275,162,105,81),such that x1,x2,x3 are adjacent to each other and to all vertices of M L.Then the resulting graph has smallest eigenvalueu3 and minimal valency 165,and contains the M L as an induced subgraph.By showing that M L is not 2-integrable,we will show that the number in the result of Koolen,Yang and Yang is at least 166.Moreover,by using vectors from shorter Leech lattice we will show that the McL is 4-integrable.Beside these in this thesis we study the s-integrality of some other interesting strongly regular graphs.