Let K be a field and n be a positive integer. Let Ξ=([n],E) be a simple graph, where [n]={1,β¦,n}. If S=K[x1β,β¦,xnβ,y1β,β¦,ynβ] is a polynomial ring, then the graded ideal LΞKβ(2)=(xiβxjβ+yiβyjβ:{i,j}βE(Ξ))βS, is called the Lov\'{a}sz-Saks-Schrijver
ideal, LSS-ideal for short, of Ξ with respect to K. In the
present paper, we compute a Gr\"obner basis of this ideal with respect to
lexicographic ordering induced by x1β>β―>xnβ>y1β>β―>ynβ when
Ξ=T is a tree. As a result, we show that it is independent of the choice
of the ground field K and compute the Hilbert series of
LTKβ(2). Finally, we present concrete combinatorial formulas to
obtain the Krull dimension of S/LTKβ(2) as well as lower and upper
bounds for Krull dimension.Comment: 23 pages, 1 figur