A Note on the Gutman Index of Jaco Graphs

The concept of the \emph{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, (\emph{directed graphs by definition}), and describe a recursive formula for the \emph{Gutman index} $Gut(J^*_{n+1}(x)).$ We also determine the \emph{Gutman index} for the trivial \emph{edge-joint} between Jaco graphs.


Introduction
For a general reference to notation and concepts of graph theory see [2]. The concept of the Gutman index Gut(G) of a connected undirected graph G was introduced in 1994 by Gutman [4]. It is defined to be Gut(G) = {v,u}⊆V (G) are the degree of v and u in G respectively, and d G (v, u) is the distance between v and u in G.
Clearly, if the vertices of G on n vertices are randomly labelled v 1 , v 2 , v 3 , ..., v n the definition states that Gut(G) = Worthy results are reported in Andova et.al. [1] and Dankelmann et.al. [3].
2 The Gutman Index of the Underlying Graph of a Jaco Graph, J n+1 (1), n ∈ N, n ≥ 2 The infinite Jaco graph (order 1 ) was introduced by Kok et.al. [5,6], and is defined by The graph has four fundamental properties which are; V (J ∞ (1)) = {v i |i ∈ N} and, if v j is the head of an edge (arc) then the tail is always a vertex v i , i < j and, if v k , for smallest k ∈ N is a tail vertex then all vertices v ℓ , k < ℓ < j are tails of arcs to v j and finally, the degree of vertex k is d(v k ) = k. The family of finite directed graphs are those limited to n ∈ N vertices by lobbing off all vertices (and edges arcing to vertices) v t , t > n. Hence, We denote the underlying graph by J * n (1). We now provide a recursive formula of the Gutman index Gut(J * n+1 (1)) in terms of Gut(J * n (1)). Theorem 2.1. For the underlying graph J * n (1) of a finitie Jaco Graph J n (1), n ∈ N, n ≥ 2 with Jaconian vertex v i we have that recursively: Proof. Consider the underlying Jaco graph, J * n (1), n ∈ N, n ≥ 2 with prime Jaconian vertex v i . Now consider J * n+1 (1). From the definition of a Jaco graph the extension from J * n (1) to J * n+1 (1) added the vertex v n+1 and the edges v i+1 v n+1 , v i+2 v n+1 , ..., v n v n+1 .
Step 1: Consider any ordered pair of vertices By applying the definition of the Gutman index to these of vertices we have the term: By applying this step ∀v k , 1 ≤ k ≤ i − 1, and ∀v q , k + 1 ≤ q ≤ i with k < q we obtain: Step 2: Consider any vertex v k , 1 ≤ k ≤ i and any other vertex v t , i + 1 ≤ t ≤ n. By applying the definition of the Gutman index to this pair of vertices we have the term: By applying this step ∀v k , 1 ≤ k ≤ i and ∀v t , i + 1 ≤ t ≤ n, we obtain: Step 3: Consider any two distinct vertices v t , v q , i + 1 ≤ t ≤ n − 1, and t + 1 ≤ q ≤ n. By applying the definition of the Gutman index to this pair of vertices we have the term: By applying this step ∀v t , i + 1 ≤ t ≤ n − 1 and ∀v q , t + 1 ≤ q ≤ n, we obtain: Step 4: Consider any vertex v k , 1 ≤ k ≤ i and the vertex v n+1 . By applying the definition of the Gutman index to this pair of vertices we have the term: By applying this step ∀v k , 1 ≤ k ≤ i we obtain: Step 5: Consider any vertex v t , i + 1 ≤ t ≤ n and the vertex v n+1 . By applying the definition of the Gutman index to this pair of vertices we have the term: By applying this step ∀v t , i + 1 ≤ t ≤ n we obtain: Final summation step: Adding Steps 1 to 5 and noting that: , provides the result: 3 The Gutman Index of the Edge-joint between J * n (1), n ∈ N and J * m (1), m ∈ N The concept of an edge-joint between two simple undirected graphs G and H is defined below.
Definition 3.1. The edge-joint of two simple undirected graphs G and H is the graph obtained by linking the edge vu | v∈V (G),u∈V (H) , and denoted G vu H.