Ordering Acyclic Connected Structures of Trees Having Greatest Degree-Based Invariants

Being building block of data sciences, link prediction plays a vital role in revealing the hidden mechanisms that lead the networking dynamics. Since many techniques depending in vertex similarity and edge features were put forward to rule out many well-known link prediction challenges, many problems are still there just because of unique formulation characteristics of sparse networks. In this study, we applied some graph transformations and several inequalities to determine the greatest value of first and second Zagreb invariant, SK and SK1 invariants, for acyclic connected structures of given order, diameter, and pendant vertices. Also, we determined the corresponding extremal acyclic connected structures for these topological indices and provide an ordering (with 5 members) giving a sequence of acyclic connected structures having these indices from greatest in decreasing order.


Introduction
e process of exploring the junctions and connections of a tree-like network is called network topology determination, where chemical compounds' entities of a complicated chemical system are represented by vertices in acyclic graphic structures.
is area of research is a well growing idea in investigation of dynamic tree-like networks because of its wide range of continuously spanning applications in different emerging fields of research. Devotion of a big amount of exploring material in literature to tree-like networks has attached great importance to acyclic graphic structures. Main idea behind consideration of tree-like network is that very often the targeting approach to trees can further be implemented or its extension can be applied to study more general and advanced networking structures. Aim of this work is to provide readers a technique to guess behavior of chemical invariants for complicated network by an easiest one tree-like network.
In this study, the term "graph" will always indicate a simple, finite, and undirected graph. In theoretical chemistry, topological indices are often used within the development of the two well-known relationships termed as quantitative structure property [1]. ese descriptors are used to build the mathematical basis for relationship between molecular structure and physico-chemical properties.
ere are several topological indices exist in the literature. In a graph K, V(K) and E(K) are the sets of vertices and edges, respectively. Let d K (s) denote the degree of a vertex s.
First time topological indices were used by Wiener launched them as, the Wiener invariant. Once having the favorable result of Wiener invariant, many other vertex degree-and edge degree-dependent invariants were proposed by several researchers, see details [2][3][4] that are given as follows: (1) e most studied and most applied index among all topological indices is the Randić index, defined by Randić [5]: (2) In 2004, Miličević et al. [6] redeveloped the Zagreb invariants using edge degrees and defined the 1 st and 2 n d reformulated Zagreb indices as follows: (3) Here, d f � deg(f) given by sum of degrees of end points of edge f decreased by 2 and f ∼ l shows that lines f and l are sharing a common node in K. Moreover, the extreme values of EM 1 (K) and EM 2 (K) were represented in [7,8].
In [9], authors put forward new graphic invariants defined below: In 2012, Xu et al. [10] established some graph transformations that maximize or minimize the multiplicative sum Zagreb index of graphs and used these graph transformations to determine the extremal graphs from tree, unicyclic, and bicyclic graphs.
Two years later, Ji et al. [8] extended the work of Xu et al. [10] for 1 st reformulated Zagreb index. Shirdel et al. [11] put forward the hyper-Zagreb index which is a degree-based topological index given by In 2017, Gao et al. [12] used the same graph transformations as given in [8] to compute the similar results as computed in [8] but for the hyper-Zagreb index. e eccentricity ecc(s) of s ∈ K is the farthest distance from s to any other vertex, i.e., ecc(s) � max t∈V(K) d (s, t).
e value of the maximum eccentricity in a graph K is called the diameter of K, and it is denoted by diam(K). Two vertices s, t ∈ V(K) are the diametral vertices of K, for which the distance between the vertices s and t is equal to diam(K) and the smallest path between vertices s and t is the diametral path. A path denoted as P m contains m number of vertices. A caterpillar, that is a tree of order 3 or more, holds the property removal of whose pendant vertices generate a path. Mahapatraa et al. [13] determined a new technique of finding link prediction called RSM index; idea behind this motivation is to increase the users on a network. Shang et al. [14] proposed the model of networks that provide a common explanation for community of regular and acyclic networks. Shang et al. [15] put forward the method taking into account heterogeneity of networks and performed in a better way than the existing link prediction algorithms. Huge collection of bounds is evaluated for acyclic and general graphic structures via Zagreb group invariants. Borovicanin et al. [16], in an attempt to take overview of existing literature regarding lower and upper bounds of Zagreb invariants, provided the readers with a broad survey of such well-known estimates. Noureen et al. [17] evaluated the maximum values of Zagreb invariants for acyclic chemical structures with certain parameters of segments and branching nodes. Ali et al. [17] provided readers with a big collection of results regarding largest and smallest values for the invariant 0 R α , taking into consideration already explored results for certain values of α, e.g., α � − 2, − 1, − (1/2), 2, 3. For further notations related to graph theory, we refer [18], and for networking dynamics, we refer [19,20].
Plan of work and methodology of this work are looking at the behavior (increase or decrease) of first two Zagreb invariants after swapping certain lines from one node to other. Increase in the value of these invariants lead us to acyclic tree-like structures with biggest value of aforementioned invariants. During this increase, these operations of swapping lines enabled us to give an ordering of tree-like structure having first, second, till fifth maximum value of considered invariants.
□ Lemma 3. Let τ 3 (K) � K ′ be an acyclic connected graph obtained from K by applying τ 3 − transform, as depicted in Figure 3, Proof. Like previous lemma and by definition of M 1 (K), we obtain

Complexity
Hence, the proof is complete.
□ Lemma 4. Let τ 4 (K) � K ′ be the graph derived from K after applying τ 4 − transform on K, as shown in Figure 4. For any s > r − 1, we have Proof.
, and by definition of M 1 (K), we have □ 2.1. Acyclic Connected Structures with Greatest M 1 Invariant. First, we identify the extremal graphs among acyclic connected graphs or (trees) with 1 st Zagreb index and provide an ordering of these trees from greatest in decreasing order for 1 st Zagreb index, see Figure 5.
e star K 1,p− 1 is a unique acyclic connected graph of diameter two, which, by Corollary 1, attains the maximum value of M 1 (K). Another maximum value of M 1 (K) reaches for S p,p− 2 � BL(p − 3, 1), which maximizes M 1 (K) in the set of acyclic connected graphs having diameter three. e next maximum values are attained by BL(p − 4, 2), which is corresponding to BL(p − 3, 1) for p � 5 and by BL(p − 5, 3) in the class of acyclic connected graphs of diameter three, and the maximum of M 1 (K) is obtained by S p,p− 3 in the class of acyclic connected graphs of diameter four. We have Since this inequality indicates that we can derive BL(p − 4, 2) from S p,p− 3 by a τ 1 − transform, it leads that, for every p ≥ 6, the acyclic connected graphs holding maximum values of M 1 (K) are K 1,p− 1 , BL(p − 3, 1) and BL(p − 4, 2). Next, we contrast M 1 (BL(p − 5, 3)) with M 1 (S p,p− 3 ) to get the 4 th term in the required series. We have which means that M 1 (BL(p − 5, 3)) < M 1 (S p,p− 3 ), for every p > 8.
Theorem 3. Let T be an acyclic connected graph having p ≥ 5 nodes and t pendant nodes, where 3 ≤ t ≤ p − 2. en, attains equality by the graph T � S p,t .
Proof. First, we prove for vertex x of degree 1, attached to a node y; we have bound attained by the graph T � S p,t and d(y) � t. We note that there exists a vertex z 0 ∈ (N(y)/ x { }) of degree d(z 0 ) ≥ 2 since in the other way T is a star having center y and t � p − 1 pendant nodes, which conflict with the theorem. We obtain  8 Complexity (26) Equality attains, by d(y) � t, a vertex adjacent to y is of degree two, and the remaining nodes are pendants, i.e., T � S p,t , and there is a vertex of S p,t of degree t to which x is adjacent.
Using induction technique for p, for p � 5, we have t � 3 and S 5,3 � BL(1, 2), as shown in Figure 5, which is the only one acyclic connected graph with order five and three pendant nodes. Let p ≥ 6, and assume that, for every acyclic connected graph of order p − 1 and with t pendant nodes, the theorem is true, where 3 ≤ t ≤ p − 3. For end node x linked with y, now investigation of two subcases is done: (a) degree of y is 2 and (b) degree of y is at least 3.
(a) Here the unique node z attached to y has d(z) ≥ 2, which means

(28)
In this situation, T − x has t pendant nodes. Using induction, for t ≤ p − 3, we get M 1 (T − x) ≤ M 1 (S p− 1,t ), which attains equality by the graph T − x � S p− 1,t . In this situation, and equality is attained by the graph T − x � S p− 1,t and d(y) � d(z) � 2, i.e., T � S p,t . If t � p − 2, order of T − x is p − 1, having p − 2 vertices of degree one, i.e., T − x is the star graph of order p deducing T � S p,p− 2 � S p,t .
(b) e graph T − x consists of p − 1 nodes and t − 1 pendant nodes if d(y) ≥ 3. en, by this property and using induction for T − x, we obtain Equality attained by the graph is T − x � S p− 1,t− 1 and d(y) � t, i.e., T � S p,t .

Second Zagreb Invariant and Graph Transformations
In this section, we make use of some graph transformations introduced by Tomescu et al. [21] to compute the 2 n d Zagreb index for acyclic connected graphs of given diameter, order. and pendant vertices. ese graph transformations are described in Section 3.

Lemma 5.
Let τ 1 (K) � K ′ be an acyclic connected graph derived from K by τ 1 − transform, as depicted in Figure 1; then, for any s, t ≥ 1.
Hence, the result holds. □ Lemma 7. Let τ 3 (K) � K ′ be an acyclic connected graph obtained from K by applying τ 3 − transform, as depicted in Figure 3, Proof. Like previous lemma and by definition of M 2 (K), we obtain 10 Complexity Hence, the proof is complete.
□ Lemma 8. Let τ 4 (K) be the graph derived from K after applying τ 4 − transform on K, as shown in Figure 4. For any s > r + 1, we have , and by definition of M 2 (K), we have

Complexity
If d K (x 1 , z 1 ) � 1, then □ 3.1. Acyclic Connected Graphs with Greatest M 2 . First, we identify the extremal graphic structures among acyclic connected graphs with 2 n d Zagreb index and provide an ordering of these acyclic connected graphs from greatest in decreasing order for 2 n d Zagreb invariant (see Figure 6). Proof. Applying τ 1 − transform in Lemma 5 at nodes not related to diametral path of T, we conclude that, between p − vertex acyclic connected graphs T of diameter d, the maximum of M 2 (T) achieves exactly in the set of multistars ML(p 1 , p 2 , . . . , p d− 1 ). After applying transformations explained in Lemmas 6, 7, and 8, we deduce that maximum of M 2 (T) attains only for p 1 � p − d, p 2 � p 3 � · · · � 0, and p d− 1 � 1, i.e., for S p,p− d+1 .