Gutman Index and Detour Gutman Index of Pseudo-Regular Graphs

The Gutman index of a connected graph G is defined as Gut(G) = ∑u ̸ =V d(u)d(V)d(u, V), where d(u) and d(V) are the degree of the vertices u and V and d(u, V) is the distance between vertices u and V. The Detour Gutman index of a connected graph G is defined as Gut(G) = ∑u ̸ =V d(u)d(V)D(u, V), whereD(u, V) is the longest distance between vertices u and V. In this paper, the Gutman index and the Detour Gutman index of pseudo-regular graphs are determined.


Introduction
All graphs considered in this paper are simple, connected, and finite.A graph is a collection of points and lines connecting a subset of them.The points and lines of a graph  are also called vertices and edges of the graph and are denoted by () and (), respectively.For , V ∈ (), the distance between  and V in , denoted by (, V), is the length of the shortest (, V)-path in .The number of vertices of  adjacent to a given vertex V is the degree of this vertex and it is denoted by (V).
A topological index is a real number related to a graph.It does not depend on the labeling or pictorial representation of a graph.The Wiener index of , () is defined as () = ∑  ̸ =V (, V), where the sum is taken through all unordered pairs of vertices of .Wiener index was introduced by Wiener, as an aid to determine the boiling point of Paraffin [1].It is related to boiling point, heat of evaporation, heat of formation, chromatographic relation times, surface tension, vapour pressure, partition coefficients, total electron energy of polymers, ultrasonic sound velocity, internal energy, and so on [2].For this reason, Wiener index is widely studied by chemists.
In [3], Gutman defined the modified Schultz index, which is known as the Gutman index as a kind of a vertex-valencyweighted sum of the distance between all pair of vertices in a graph.Gutman revealed that, in the case of acyclic structures, the index is closely related to the Wiener index and reflects precisely the same structured features of a molecular as the Wiener index does.
The Gutman index of G, denoted by Gut(), is defined as Gut() = ∑ {,V}⊆()   ()  (V)  (, V) = (1/2) ∑ ,V∈()   ()  (V)  (, V) with the summation runs over all pair of vertices of .Dankelmann et al. [4] presented an upper bound for the Gutman index and also established the relation between the Edge-Wiener index and Gutman index of graphs.Chen and Liu [5] studied the maximal and minimal Gutman index of unicyclic graphs, and they also determined the minimal Gutman index of bicyclic graphs [6].
For any acyclic connected graph, the shortest distance (, V) is the same as the longest distance (, V).But for cyclic structured graphs, the shortest distance (, V) naturally is not equal to the longest distance (, V).In this paper, the new index which is Gutman index of graphs with respect to detour distance [7,8] is considered; this is named as Detour Gutman index which is defined as A graph  is called pseudo-regular graph [9,10] if every vertex of  has equal average degree.The main goal of this paper is to find the exact formula for Gutman index and Detour Gutman index of pseudo-regular graphs.

Pseudo-Regular Graphs
Let  = (, ) be a simple, connected, undirected graph with  vertices and  edges.For any vertex V  ∈ , the degree of V  is the number of edges incident on V  .It is denoted by   or (V  ).A graph  is called regular if every vertex of  has equal degree.A bipartite graph is called semiregular if each vertex in the same part of a bipartition has the same degree.The 2degree V  [9] is the sum of the degree of the vertices adjacent to V  and denoted by   [11].The average degree of V  is defined as   /  .For any vertex V  ∈ , the average degree of V  is also denoted by (V  ) =   /  .
A graph  is called pseudo-regular graph [9] if every vertex of  has equal average degree and () = (1/) ∑ V∈() () is the average neighbor degree number of the graph .
A graph is said to be -regular if all its vertices are of equal degree .Every regular graph is a pseudo-regular graph [10].But the pseudo-regular graph need not be a regular graph.Pseudo-regular graph is shown in Figures 1 and 2.
In Figure 1, there are 14 vertices of degree 1, 7 vertices of degree 3, and 1 vertex of degree 7.So totally there are 22 vertices in the graph.Average degree of vertices of degree 1 is equal to 3/1 = 3.Average degree of vertices of degree 2 is equal to 9/3 = 3.Average degree of vertices of degree 7 is equal to 21/7 = 3.Therefore, average degree of each vertex is 3. Hence, it is a pseudo-regular graph.In Figure 2, average degree of each vertex is 5. Hence, the graph in Figure 2 is also a pseudoregular graph.
The relevance of pseudo-regular graph for the theory of nanomolecules and nanostructure should become evident from the following.There exist polyhedral (planar, 3-connected) graphs and infinite periodic planar graphs belonging to the family of the pseudo-regular graphs.Among polyhedral, the deltoidal hexecontahedron possesses pseudoregular property.The deltoidal hexecontahedron is a Catalan polyhedron with 60 deltoid faces, 120 edges, and 62 vertices with degrees 3, 4, and 5, and average degree of its vertices is 4.
In this paper, the Gutman index and Detour Gutman index for three types of pseudo-regular graphs are derived.For construction of pseudo-regular graphs, see [12].

Gutman Index and Detour Gutman Index of Type I Pseudo-Regular Graph
Theorem 1.For  ≥ 2, the Gutman index of type I pseudoregular graph  I is Proof.Let  =  I be the type I pseudo-regular graph. Let . .,  (−1) } be the vertex set of  and let V 0 be the central vertex of star graph  1, , {V 1 , V 2 , . . ., V  } are pendant vertices of  1, , where  =  2 −+1, and the (−1) pendant vertices { 1 ,  2 , . . .,  (−1) } are attached with The pseudoregular graph of type I for  = 3 is shown in Figure 3.

The Gutman index of any graph 𝐺 is given by Gut
(2)

Gutman Index and Detour Gutman Index of Type II Pseudo-Regular Graph
Theorem 2. For  ≥ 3, the Gutman index of type II Pseudoregular graph  II is Proof.Let  =  II be a type II pseudo-regular graph. Let . .,  (−3) } be the vertex set of  and V 0 as the central vertex of wheel graph   and {V 1 , V 2 , . . ., V  } are vertices of cycle   in the clockwise direction and { 1 ,  2 , . . .,  (−3) } are the pendant vertices joined to every vertex in the cycle except the central vertex, where  =  2 − 3 + 3.
The pseudo-regular graph of type II for  = 5 is shown in Figure 4.
The Gutman index of  is given by Gut Theorem 3.For  ≥ 3, the Detour Gutman index of type II pseudo-regular graph  II is Proof.Let  =  II be a type II pseudo-regular graph.

Gutman Index of Pseudo-Regular Graph of Type III
Theorem 4. For  ≥ 5, the Gutman index of type III pseudoregular graph  III is Proof.Let  =  III be a type III pseudo-regular graph. Let . .,  (−5) ,  1 ,  2 , . . .,   } be the vertex set of  and V 0 as the central vertex of wheel graph   , where  =  2 − 3 + 1. Let The pseudo-regular graph of type III for  = 6 is shown in Figure 5.
The Gutman index of  is given by Gut     Proof.Let  =  III be a type III pseudo-regular graph.

Conclusion
In this paper, we clearly determined the Gutman index of pseudo-regular graphs and also determined the Detour Gutman index of pseudo-regular graphs.

Figure 3 :
Figure 3: Pseudo-regular graph of type I at  = 3. v