AnApproach to the Extremal Inverse Degree Index for Families of Graphs with Transformation Effect

Department of Mathematics, COMSATS University Islamabad, Lahore Campus, Lahore 54000, Pakistan Faculty of King Abdulaziz City for Science and Technology (KACST) Riyadh, Riyath, Saudi Arabia IT Programs Center, Faculty of IT Department, Institute of Public Administration, Riyadh 11141, Saudi Arabia Department of Natural and Applied Sciences, Faculty of Community College, Majmaah University, Majmaah 11952, Saudi Arabia


Introduction and Preliminary Results
Graph theory has many applications in chemistry, physics, computer sciences, and other applied sciences. Topological indices are graph invariants used to study the topology of graphs. Along with the computer networks, graph theory considers as a powerful tool in other areas of research, such as in coding theory, database management system, circuit design, secret sharing schemes, and theoretical chemistry [1]. Cheminformatics is the combination of technology, graph theory, and chemistry. It develops a relationship between structure of organic substances and their physiochemical properties through some useful graph invariants with the help of their associated molecular graph. e molecular graph is the combination of vertices and edges which are representatives of atoms and bonds between atoms of corresponding substance, respectively. eoretical study of underlying chemical structure by some useful graph invariants is an attractive area of research in mathematical chemistry due to its effective applications in the QSAR/ QSPR investigation [2,3]. Topological indices among these invariants have special place and used to estimate the physiochemical properties of chemical compound. A topological index can be considered as a function which maps a graph to a real number. roughout this work, we used standard notations, G � G(V, E) for graph, V(G) set of vertices, E(G) the set of edges, d v i degree of vertex v i (the number of edges incident to v i ), Δ and δ be the maximum and minimum degrees of fully connected vertices, vertices with degree one are pendent vertices, and path attached with fully connected vertices taken as a pendent paths.
In the last five decades, after the Wiener index, many topological indices had been introduced. Probably, the Randić connectivity index [4] is one of the best predictive invariants among these topological indices. e accuracy in predictability of indices is the main interest of researchers which leads them to purpose a new topological index. e zeroth-order general Randić index 0 R α (G) � u∈V(G) d α u was conceived by Li and Zheng in their work [5]. [6] and others [7][8][9][10] characterize 0 R α for different values of α. For α � − 1, 0 R − 1 � ID is modified total adjacency index or inverse degree, first appears in the conjecture over computer program Graffiti [11]. e for graphs without isolated vertices are well discussed in [12,13]. Extremal characterization and bonds of ID(G) also discussed at some extent in [14][15][16][17][18]. For more detail, one can review survey [19].
In this work, we investigated the effect of transformations over families of graphs for ID and established inequalities for these transformed graphs. Graph transformations are very important in chemistry, computer designing, and animations. Moreover, we determined the exact value of ID for some major families of graphs under the effect of transformations over pendent paths.

Results and Discussion
In this section, we present some transformations over pendent paths. ese have solid effect over increase and decrease of ID(G). rough out this work, we considered n 0 -vertex connected graph G 0 . G l k be the graph G 0 with k pendent paths of length l ≥ 1 having order n � n 0 + kl with degree sequence d 1 � δ ≤ d 2 ≤ d 3 ≤ · · · ≤ Δ + 1.

Graph Transformations. Let
be subgraph obtained by removing edges of E(G ′ ), and G 1 ′ � G − V ′ (G) be the subgraph obtained by deleting vertices set V ′ (G) ⊂ V(G) along with their incident edges. We give following transformations using these techniques which have solid effect on ID(G).

2.1.1.
Transformation . . , u l− 1 j u l j be the pendent paths attached with fully connected vertex w j of G 0 forms G l k . en, (3) Figure 1 depicts successive application of transformation A as A i , i � 1, 2, 3, . . . , l − 1.

2.1.2.
Transformation B. Let w j ∈ V(G 0 ), d w j ≥ 3, j � 1, 2, 3, . . . , k ≤ n and w j u 1 j , w j u 1 j , w j u 3 j , . . . , w j u l− i j be the leafs attached with fully connected vertex w j of G. en, for fixed vertex w 1 , Theorem 1. Let G 0 be the graph of order n 1 with maximum degrees Δ and minimum δ. en, Proof. Let G l k be the graph of order n � n 1 + kl, minimum degree δ, and maximum degree Δ + 1. G l k is the composition of G 0 and k pendent paths of length l. In G l k , there are at least k vertices of degree 1 and k(l − 1) having 2 and n 1 vertices e transformation A transforms k vertices from degree 2 to 1 and another k vertices from d s + 1 to d s + 2 which have an effect in ID as So, from equations (6) and (7), we have Replace d v s with minimum degree δ. It maximizes the term k 2 Journal of Chemistry e transformation B shown in Figure 2 decreases the degree of one vertex and makes the same increase into the degree of fixed selected vertex: So, from equations (6) and (11), we get to maximize the fraction involved in above expression replace d v s , 0 ≤ s ≤ k with δ, and we get e equation (13) implies For main results related to the transformation A j i shown in Figure 3, we need to prove Propositions 1 and 2.
Proof. (1) If transformation A applied on pendent path attached with vertex w j of G having degree η + ζ. e degree of vertex w j increased by one with change of vertex having degree 2 to leaf attached to w j . is change has effect on ID in the following way. Let g(η, ζ) � 1/(η + ζ). en, implies g(η, ζ) + ζ ≥ g(η, ζ − 1).
(2) e 2nd part of this preposition is related to the effect of transformation A j i shown in Figure 3: Journal of Chemistry is fraction is nonnegative for all η ≥ 1 which implies that g(η) ≥ f(η). □ Theorem 2. Let G be the graph of order n having p pendent vertices. G l k is the graph having k pendent paths attached to the fully connected vertices with maximum degree of a vertex Δ + 1. en, for 0 ≤ i ≤ l − 1 and α ≤ β, Proof. Let G l k be the graph with order n � n + kl, minimum degree δ, and maximum degree Δ + 1. Using the fact of A j i over G l k , we get for 1 ≤ d r , d s ≤ Δ ≤ n − 1 and 0 ≤ i ≤ l − 1, 0 ≤ j ≤ k − 1. en, for α ≤ β, So, by using Proposition 1 and replacing Δ with d v r and δ with d v s , it is clear that Δ minimizes the positive terms and δ maximizes the negative term. After simplification, we get It is clear that for α ≤ β, the nominator is a negative number and denominator is positive which implies that us, for β ≥ α, □ In the following theorem, we determined bonds of ID for graph G l k under the effect of transformation A j i by using Propositions 1 and 2.
Theorem 3. Let G be the graph of order n having p pendent vertices. G l k is the graph with maximum degree Δ + 1 having order n + kl with k ≤ n − p of pendent paths of length l. en, Equality holds for r-regular graph with i � 0, j � 0, and k � 1. And equality holds if under consideration graph is r-regular with k � n − p pendent paths of length l and i � l − 1, j � k − 1.
Proof. Let G be the graph having order n ≥ 3 with 0 ≤ p ≤ n − 1 pendent vertices with minimum degree δ and maximum degree Δ. G l k is the graph with maximum degree Δ + 1 and maximum number of pendent paths k � n − p of length l. en, by using equation (20), where 1 ≤ d r , d s ≤ Δ ≤ n − 1, and 0 ≤ i ≤ l − 1, 0 ≤ j ≤ k − 1. e order of G l k is fixed. So, the increase in pendent paths causes to decrease their lengths l.
is fact increases the number of pendent paths and decreases the vertices of degree two. So, Proposition 2 clears that ID(A j i (G l k )) increases with the increase in k. It is clear from eorems 1 and 2 and Propositions 1 and 2 that the least value of ID(A j i (G l k )) was obtained by setting i, j � 0, d r � d s � Δ, and k � 1: After simplification, we get and equality holds in (28) for r-regular graph with the k � 1 pendent path of length l and i � 0, j � 0. Now again from (20), setting i � l − 1, j � k − 1, d r � d s � δ, k � n − p and using Proposition 1 and eorems 1, 2, we get maximal value of ID(A j i (G l k )) as  Journal of Chemistry 5 After simplification, we get maximal value of in which equality holds for r-regular graph with k � n − p pendent paths of length l and i � l − 1, j � k − 1. Inequalities (29) and (31) complete the proof.
□ Theorem 4. Let G be the graph without pendent vertices and G l k for k > 1 be the graph with maximum degree Δ + 1. en, Equality holds for r-regular graph with k pendent paths of length l and i � 0, j � 0.
Proof. Let G be the graph of order n without pendent vertices having minimum degree δ and maximum degree Δ. G l k be the graph with maximum degree Δ + 1 and k ≥ 1 be the count of pendent paths of length l. en, by using (20), Using Propositions 1 and 2 and setting i, j � 0, d r � d s � Δ, we get least value of ID(A j i (G l k )) as ID(A 0 0 (G l k )): After simplification, we get minimal value as Equality for equation (3) holds for r-regular graph with k pendent paths of length l and i � 0, j � 0. en, for Proof. Let G be the Δ-regular graph and k be the count of pendent paths of length l. en, G l k is the graph with maximum degree Δ + 1. en, for 0 After simplification, we get required result: □ In Corollary 1, we determined exact values of ID for unicyclic graphs with k pendent paths of length l under transformation A j i . Figure 4(a) depicts transformed graph A j i (C l n,k ) for i � j � 1, k � 2 with fixed vertex w 1 and Figure 4(b) with fixed vertex w 2 .

Conclusions
Topological indices and graph transformations play a significant role in modern chemistry and computer networks. It is an interesting problem to determine the bonds of the topological index for different families of graphs [5,6,9,16]. In this work, we give graph transformations, A, B, and A j i for variable values of i and j over pendent paths attached with the fully connected vertices of graphs and characterized ID for these transformed graphs. At first, we determined the effect of transformations A and B over increase and decrease of ID individually. en, we established result for A j i for arbitrary values of i and j which provides moving graphs such as animation. We also determined the exact result for Δ-regular graph under transformation effect. Moreover, we computed the exact formula for the family of unicyclic graphs with pendent paths under the action of transformation as an application of proved results.

Data Availability
No data were used to support this study. Disclosure e paper has not been published elsewhere, and it will not be submitted anywhere else for publication.

Conflicts of Interest
e authors declare that they have no conflicts of interest.