Detour-Cluj matrix and derived invariants

Detour-variant, CJ Delta(u), of the recently proposed Cluj matrix, CJD(u), is defined and exemplified. Cluj indices built up on the two variants of the Cluj matrix are evaluated and compared, for selected sets of graphs. A new graph-theoretical local property is defined, namely, the ''internal ending of all longest paths joining a vertex i, with deg(i) > 1, with all the remaining vertices in the graph'', as the property of the vertex i (called an internal endpoint) to have all its entries in CJ Delta(u) equal to 1. Classes of graphs possessing a minimal CJ Delta(p) value (i.e., in which all vertices are internal endpoints) have been identified. Correlating tests of the invariants calculated on the two variants of Cluj matrices are also performed.