Influence of the boundary on the connective constant of branching structures

It is shown that the connective constant of a branching structure is modified if account is taken of the presence of a boundary containing a finite fraction of the vertices of the system. In particular, for a Cayley branch with branching ratio m, the calculation of the self-avoiding walk generating function shows that the connective constant is μ = m, whereas the corresponding Bethe lattice result is μ = m. As a second example, a triangular cactus tree is studied, giving μ = 12(42 + 2 +2) in contrast to the value μ = 1 + 3 for the corresponding infinite graph.