Closed trail distance in a biconnected graph

Graphs describe and represent many complex structures in the field of social networks, biological, chemical, industrial and transport systems, and others. These graphs are not only connected but often also k-connected (or at least part of them). Different metrics are used to determine the distance between two nodes in the graph. In this article, we propose a novel metric that takes into account the higher degree of connectivity on the part of the graph (for example, biconnected fullerene graphs and fulleroids). Designed metric reflects the cyclical interdependencies among the nodes of the graph. Moreover, a new component model is derived, and the examples of various types of graphs are presented.