A boundary transitive graph \(X\) is a locally finite connected graph with infinitely many ends whose automorphism group is noncompact and transitive on the space of ends. One shows that a closed noncompact transitive group \(G\) of automorphisms of \(X\) is of split rank one. It follows that each positive multiplicative cocycle of the action of \(G\) on the boundary of \(X\) is cohomologous to a power of the Radon-Nikodym derivative cocycle, relative to the action of \(G\) on a \(K\)-invariant measure on the boundary, where \(K= G_x\) \((x \in X)\). When \(X\) is a semi-homogeneous tree one determines the irreducible representations of the algebra of operators on \(L^2(X)\) commuting with the \(G\)-action and preserving the space of functions with finite support. These representations are parametrized by a complex number \(z\). They are two-dimensional with the exception of a discrete set of values of \(z\) for which they are one-dimensional. These results follow from the study of random walks on \(X\), their Martin boundaries and their Martin kernels, which are computed using the above property of the positive multiplicative cocycles.