Constructing 3-connected graphs with coinciding path layer matrices (Q2713919)

Let \(G=(V,E)\) be a graph, let \(\lambda_{ij}\) be the number of vertices at distance \(j\) from the vertex \(v_i\), and let \(\tau_{ij}\) be the number of simple paths of length \(j\) starting at \(v_i\). The layer matrix of \(\lambda(G)\) and the path layer matrix of \(\tau(G)\) extend the concepts of distance degree sequence and path degree sequence of the graph \(G\). These matrices are useful in chemical applications.NEWLINENEWLINENEWLINEThe main result consists in constructing several infinite families of non-isomorphic 2- and 3-connected graphs with the same path layer matrices. In particular, some open questions have been answered.