Minimum atom-bond sum-connectivity index of unicyclic graphs with maximum degree

Let G be a graph with edge set E ( G ) . Denote by d u the degree of a vertex u in G . The atom-bond sum-connectivity (ABS) index of G is deﬁned as ABS ( G ) = (cid:80) xy ∈ E ( G ) (cid:112) ( d x + d y − 2) / ( d x + d y ) . In this article, we determine the minimum possible value of the ABS index of unicyclic graphs of order n and maximum degree ∆ such that 3 ≤ ∆ ≤ n − 2 . All the graphs that attain the obtained minimum value are also characterized.


Introduction
Consider a connected and simple graph G, represented as G = (V, E), with order n and size m.The graph G a c-cyclic graph when m = n + c − 1.Specifically, when c = 1, G is referred to as a unicyclic graph.Let P n and S n be respectively the path and star graphs with n vertices.The set of neighbors of a vertex y in G is represented by N G (y) (or simply by N (y)) and the degree of y in G is represented by d y (G) (or simply by d y ).The symbols δ = δ(G) and ∆ = ∆(G) are used to represent the minimum and maximum degrees of G, respectively.A vertex of degree 1 in G is referred to as a leaf or an end-vertex, while a vertex adjacent to a leaf is known as a stem.On the other hand, a strong stem is a stem that is adjacent to two or more leaves.A stem having at most one non-leaf neighbor is called an end stem.
Consider two different subsets U and V of V (G).A path P in G that begins at a vertex in U and ends at a vertex in V such that all other vertices of P do not belong to either of the sets U and V is known as an (U, V)-path.If V = {v}, we write (U, v)-path instead of (U, V)-path.The length of a shortest (U, V)-path is referred to as the distance between the sets U and V in the graph G, denoted as d G (U, V).
Chemical graph theory [22] is a fundamental branch of mathematical chemistry, which focuses on representing and analyzing chemical structures using mathematical models.In chemical graph theory, topological indices play a crucial role in establishing connections between the structures of molecules and their properties related to the development of computer-aided drug designs [10,16].Numerous topological indices based on vertex degrees have been discussed in [12,15].In this paper, we are concerned with the atom-bond sum-connectivity (ABS) index, which is a recently introduced topological index [5].The ABS index of G is defined as follows: In [5], various basic mathematical properties of the ABS index were established.Ali et al. [6] examined the chemical applicability of the ABS index and identified its extremum values for unicyclic graphs.Gowtham and Gutman [14] formulated several inequalities between the ABS index and another existing connectivity index.In [8,9], Alraqad et al. studied some extremal problems related to this index for certain classes of graphs.Noureen and Ali [20] solved the problem of determining the largest ABS index of trees of a given order with a fixed number of leaves.In [19], we determined the first four smallest values of the ABS index of unicyclic graphs of a given order with a specific girth and identified the graphs that achieve the obtained extremal values.For more details about the ABS index and its applications in the field of chemistry, additional information can be found in the references [1-3, 7, 13, 18, 19, 25].
Recently, Hussain et al. [17] determined some sharp bounds for the ABS index in terms of other graph invariants.Among other things, a lower bound for the ABS index of trees in terms of their order and maximum degree was found in [17].In this article, we extend this lower bound of Hussain et al. [17] to the one for unicyclic graphs.
Let U n be the set of all unicyclic graphs on n ≥ 5 vertices.Denote by U n,k the subset of U n that consists of those members of U n that have a fixed girth k, 3 ≤ k ≤ n.Certainly, U n = n k=3 U n,k and U n,n = {C n }, where C n is the cycle on n vertices.Let U 1 n,n−1 be the unique unicyclic graph with n vertices and girth n − 1.Thus,  The main objective of this study is to determine the minimum possible value of the ABS index of unicyclic graphs of order n and maximum degree ∆ such that 3 ≤ ∆ ≤ n − 2, and characterize all the graphs that attain this minimum value.To prove some results of this paper, we utilize the following theorems: Theorem 1.2.[19] Consider a non-trivial connected graph H with 2 distinct vertices x and y such that both x and y have at least two neighbors.Additionally, suppose that when y (or x) has precisely two neighbors, the sum of these neighbors is at most 7 in H. Create a new graph G from H by adding two paths P and Q, to the vertices x and y, respectively, where P := xx 1 . . .x a and Q := yy 1 . . .y b (a ≥ b ≥ 1).Now, obtain a new graph G * from G by adding an edge between y b and x 1 and removing the edge between x and x 1 .Then ABS(G) > ABS(G * ).

Main results
Throughout this section, we denote by G a unicyclic graph with n vertices, where n ≥ 5.The unique cycle of the graph G is denoted as C. Let w ∈ V (G) be the vertex with maximum degree ∆ such that its distance from the set of vertices of the cycle Hence, ABS(G) = xy∈E(G) abs w (xy).The next two results are direct consequences of Theorem 1.1 and Theorem 1.2.
Corollary 2.1.Consider a unicyclic graph G with order n and maximum degree ∆ such that 3 ≤ ∆ ≤ n − 2. If G possesses a strong stem (a stem with at least three neighbors) that is distinct from the vertex w, then there is a unicyclic graph G * with order n and maximum degree ∆ such that then ABS(G) > ABS(G * ).
Corollary 2.2.Consider a unicyclic graph G with order n and maximum degree ∆ such that 3 ≤ ∆ ≤ n − 2. Let P := y 0 y 1 . . .y k = w be the shortest path from a vertex y 0 ∈ V (C) to w (i.e., (V (C), w)-path), which may be of length zero.If G contains a vertex of degree at least three, excluding w and y 0 , then there is a unicyclic graph G * with order n and maximum degree ∆ such that ABS(G) > ABS(G * ).First, consider the case when d(y 0 ) ≥ 4. If y 0 is a stem of degree at least 3 and is different from w, then the result follows from Corollary 2.1.However, if y 0 is not a stem, let x 0 ∈ N (y 0 ) − {y 1 , y 2 , y 3 } and let T y0 be the component of G − {y 0 y 1 , y 0 y 2 , y 0 y 3 } containing y 0 .The longest path in T y0 is represented as y 0 x 0 . . .
By the definition of the ABS index, we have

It is clear that ABS(G) > ABS(G * ).
In all cases, we arrive at the desired inequality.
By Corollary 2.2 and Lemma 2.1, we may make the assumption that w is a vertex on the cycle C and it is the only vertex in G having a degree greater than two.
Contracting an edge e in a graph G gives a new graph G/e obtained from G by removing e and then merging its end vertices into a single vertex.Lemma 2.2.Consider a unicyclic graph G with order n and maximum degree Create a new graph G * obtained from G by identifying the vertices w and w 1 and adding a new pendant edge w 3 w (see Figure 2.2).It is clear that G * is a unicyclic graph with order n and ∆(G * ) = ∆.Now, let S = {ww 1 , w 1 x 1 , ww 3 }.Using the definition of the ABS index, we have Since ∆ ≥ 3, from ( 8) and ( 9), it follows that ABS(G) > ABS(G * ).
Recall the structure of the graph U k1,k2,...We are now prepared to present the proof of our main result.
Theorem 2.1.Consider a unicyclic graph G with order n and maximum degree ∆ such that 3 ≤ ∆ ≤ n − 2, then Equality is achieved if and only if G ∈ F 1 ∪ F 2 .
Proof.Let U ∆ n be the collection of unicyclic graphs with n vertices and maximum degree ∆ such that 3 ≤ ∆ ≤ n − 2. Consider G ∈ U ∆ n such that ABS (G) = min ABS (G 1 ) : G 1 ∈ U ∆ n .Consider a vertex w ∈ V (G) with maximum degree ∆.By Corollary 2.2 and Lemma 2.1, it is concluded that w ∈ V (C) and it is the only vertex with a degree exceeding two in G.With the help of Lemma 2.2 and Lemma 2.3 along with our choice of G, we conclude that G ∈ F 1 (n, ∆) ∪ F 2 (n, ∆).
k the unicyclic graph created by starting with the cycle C k and then attaching b paths to a single vertex x of C k such that the attached paths have lengths k 1 , k 2 , . . ., k b provided that b j=1 k j = n − k, see Figure 1.1.

Theorem 1 . 1 . [ 19 ]
Consider a non-trivial connected graph H with a vertex x.Create a new graph G by adding 2 pendant paths P and Q to the vertex x.The paths are defined as P := xx 1 . . .x a and Q := xy 1 . . .y b , where a ≥ b ≥ 1).Now, obtain a new graph G * from G by adding an edge between vertices x a and y 1 and removing the edge between x and y 1 .Then ABS(G) > ABS(G * ).

Lemma 2 . 3 .= 1 4 w 3 w b4 4 4 , w b4−1 4 w b4−2 4 .
Consider a unicyclic graph G with order n and maximum degree ∆ such that 4 ≤ ∆ ≤ n − 2. If w is a stem, w 3 is a leaf adjacent to w, and there exists an end vertex y with d (y, w 3 ) ≥ 4, then there is a unicyclic graph G * with order n and∆(G * ) = ∆ such that ABS(G) > ABS(G * ).Proof.Suppose that N (w) ∩ V (C) = {w 1 , w 2 }.Consider the component T w of G − E(C)containing w and within T w , there exists a longest path ww 0 i w 1 i . . .w bi i , where w i = w 0 i , for each 4 ≤ i ≤ ∆(G).In this path d w bi i for each i and d w j i = 2 otherwise.Assuming that b 4 ≥ 2. Create a new graph G * = G − w b4 4 w b4−1 as illustrated in Figure 2.3.Additionally, take S = ww 3 , w b4 4 w b4−1 It is clear that G * is a unicyclic graph with n vertices and ∆(G * ) = ∆.Using the definition of the ABS index, we have