2 5 Fe b 20 15 A note on the Gutman Index of Jaco Graphs , J n ( 1

The concept of the Gutman index, denoted Gut(G) was introduced for a connected undirected graph G. In this note we apply the concept to the underlying graphs of the family of Jaco graphs, (directed graphs by definition), and decribe a recursive formula for the Gutman index Gut(J∗ n+1(1)) in terms of the Gutman index of the underlying Jaco graph, J ∗ n (1), n ∈ N with prime Jaconian vertex vi. We also determine Gut(J ∗ n(1) v1u1 J ∗ m(1)) in terms of Gut(J ∗ n(1)) and Gut(J∗ m (1)). The aforesaid is the edge-joint, J∗ n (1) ∪ J∗ m (1) + v1u1, v1 ∈ V (J ∗ n (1)) and u1 ∈ V (J ∗ m (1)). We believe that the scope of results stemming from researching Gut(J∗ n (1) vkut J ∗ m (1)), vk ∈ V (J∗ n(1)), ut ∈ V (J ∗ m(1)) with 1 ≤ k ≤ n and 1 ≤ t ≤ m in terms of Gut(J ∗ n(1)) and Gut(J ∗ m(1)), will be worthy. We pose two open problems as well. Settling the open problems might enable us to significantly simplify the declared results.