SHARP UPPER AND LOWER BOUNDS ON THE NUMBER OF SPANNING TREES IN CARTESIAN PRODUCT OF GRAPHS

Let G(1) and G(2) be simple graphs and let n(1) = vertical bar V(G(1))vertical bar, m(1) = vertical bar E(G(1))vertical bar, n(2) vertical bar V(G(2))vertical bar and m(2) vertical bar E(G(2))vertical bar. In this paper we derive sharp upper and lower bounds for the number of spanning trees 'T in the Cartesian product G(1)square G(2) of G(1) and G(2). We show that: tau(G(1)square G(2)) >= 2((n1-1)(n2-1))/n(1)n(2)(tau(G(1))n(1))(n2+1/2)(tau(G(2))n(2))(n1+1/2) and tau(G(1)square G(2)) <= tau(G(1))tau(G(2)) [2m(1)/n(1)-1 + 2m(2)/n(2)-1]((n1-1)(n2-1)) We also characterize the graphs for which equality holds. As a by-product we derive a formula for the number of spanning trees in K-n1 square K-n2 which turns out to be n(1)(n1-2)n(2)(n2-2)(n(1) + n(2))((n1-1)(n2-1)).