AKCE International Journal of Graphs and Combinatorics
A vertex (respectively, edge) cycle stochastic function of a graph G is a labeling of vertices (respectively, edges) by a non-negative real valued function (Formula presented.) (respectively, (Formula presented.)) such that for every cycle of G, the sum of labels of its vertices (respectively, edges) is 1. The graphs where we can define such a function are called vertex cycle stochastic graphs (respectively, edge cycle stochastic graphs). In this paper, we provide a structure theorem for biconnected cycle stochastic graphs, which is extended to characterize edge cycle stochastic graphs. We also find a minimal forbidden graph characterization for biconnected vertex cycle stochastic graphs and its description for vertex cycle stochastic graphs.
Rao, S. B.; Sahoo, U. K.; and Parameswaran, V., "Cycle stochastic graphs: Structural and forbidden graph characterizations" (2020). Journal Articles. 511.