Addressing graph products and distance-regular graphs

Abstract

Graham and Pollak showed that the vertices of any connected graph G can be assigned t-tuples with entries in {0, a, b}, called addresses, such that the distance in G between any two vertices equals the number of positions in their addresses where one of the addresses equals a and the other equals b. In this paper, we are interested in determining the minimum value of such t for various families of graphs. We develop two ways to obtain this value for the Hamming graphs and present a lower bound for the triangular graphs.

Addressing Johnson graphs, complete multipartite graphs, odd cycles and other graphs

“…In this paper, we prove that N 2 pJpn, kqq ď kpn ´kq by constructing an explicit addressing of Jpn, kq with p0, 1, ˚q-words of length kpn ´kq. We answer a question from [4] and show that N 2 pJpn, 2qq " 2pn ´2q for n " 5, 6. In the case of n " 6 and k " 3, using the computer, we prove that N 2 pJp6, 3qq " 8 which is smaller than our general bound above.…”

“…Elzinga, Gregory and Vander Meulen [5] proved that the Petersen graph does not have an eigensharp addressing and found an optimal addressing of it of length 6 (one more than the lower bound (1)). Cioabȃ, Elzinga, Markiewitz, Vander Meulen and Vanderwoerd [4] gave two proofs showing that the Hamming graphs have eigensharp addressings and started the investigation of optimal addressings for the Johnson graphs. The Johnson graph Jpn, kq has as vertices all the k-subsets of the set t1, .…”

“…In the case of n " 6 and k " 3, using the computer, we prove that N 2 pJp6, 3qq " 8 which is smaller than our general bound above. The best known lower bound is N 2 pJpn, kqq ě n (see [4,Theorem 5.3]). For r ě 3, Watanabe, Ishii and Sawa [14] studied p0, 1, .…”