Some Bond Incident Degree Indices of (Molecular) Graphs with Fixed Order and Number of Cut Vertices

A bond incident degree (BID) index of a graph G is defined as 􏽐uv∈E(G)f(dG(u), dG(v)), where dG(w) denotes the degree of a vertex w of G, E(G) is the edge set ofG, andf is a real-valued symmetric function.(e choice f(dG(u), dG(v)) � aG + aG in the aforementioned formula gives the variable sum exdeg index SEIa, where a≠ 1 is any positive real number. A cut vertex of a graph G is a vertex whose removal results in a graph with more components than G has. A graph of maximum degree at most 4 is known as a molecular graph. Denote by Vn,k the class of all n-vertex graphs with k≥ 1 cut vertices and containing at least one cycle. Recently, Du and Sun [AIMSMathematics, vol. 6, pp. 607–622, 2021] characterized the graphs having the maximum value of SEIa from the set Vkn for a> 1. In the present paper, we not only characterize the graphs with the minimum value of SEIa from the set V k n for a> 1, but we also solve a more general problem concerning a special type of BID indices. As the obtained extremal graphs are molecular graphs, they remain extremal if one considers the class of all n-vertex molecular graphs with k≥ 1 cut vertices and containing at least one cycle.


Introduction
Graph invariants of the following form are known as the bond incident degree (BID) indices [1][2][3][4]: where d G (w) denotes the degree of a vertex w of the graph G, E(G) is the edge set of G, and f is a real-valued symmetric function. In this paper, we are concerned with the following type [5] of the BID indices: where i ∈ 1, 2 { }, f 1 is a strictly increasing and strictly convex function, while f 2 is a strictly decreasing and strictly concave function.
If a ≠ 1 is a positive real number, then the variable sum exdeg index SEI a of a graph G can be defined as e graph invariant SEI a was introduced in 2011 by Vukičević [6] for predicting the octanol-water partition coefficient of chemical compounds. For detail about the mathematical results on the variable sum exdeg index, we refer the interested readers to references [7][8][9][10][11][12].
A cut vertex of a graph G is a vertex whose removal results in a graph with more components than G has. An n-vertex graph is a graph of order n. Let V k n be the set of all n-vertex graphs with k ≥ 1 cut vertices and containing at least one cycle.
Nowadays, finding graphs with maximum or minimum values of some graph quantity from a given class of graphs is one of the popular problems in chemical graph theory.
Recently, Du and Sun [13] characterized the graphs with the maximum variable sum exdeg index SEI a from the set V k n when a > 1. e main motivation of the present paper comes from [13]. In this paper, we not only characterize the graphs with the minimum variable sum exdeg index SEI a from the set V k n for a > 1, but also we solve a more general problem concerning the BID indices I f 1 and I f 2 . By using the obtained general result, we also characterize the graphs with the minimum general zeroth-order Randić index 0 R α (see [14]) when α > 1, minimum multiplicative second Zagreb index Π 2 (see [15]), and minimum sum lordeg index SL (see [16][17][18] We note that equation (5) gives which is minimum in a given class of graphs if and only if Π 2 (G) is minimum in the considered class of graphs. A graph of maximum degree at most 4 is known as a molecular graph. As the obtained extremal graphs are molecular graphs, they remain extremal if one considers the class of all n-vertex molecular graphs with k ≥ 1 cut vertices and containing at least one cycle.
All the graphs considered in this paper are connected. e notation and terminology that are used in this paper but not defined here can be found in some standard graphtheoretical books, like [19,20].

Lemmas
A cut vertex of G is a vertex whose removal increases the number of components of G. Denote by V k n the class of all connected n-vertex graphs with k cut vertices and containing at least one cycle. In this section, in order to obtain the main result, we establish some preliminary lemmas.

Lemma 1.
If v 1 and v 2 are adjacent vertices of a graph G, then it holds that Proof.
e proof follows directly from the definitions of I f 1 and I f 2 . □ A cactus graph is a connected graph in which every pair of cycles has at most one vertex in common.

respectively) value among all graphs of the class V k n , then G is a cactus graph.
Proof. If G is not a cactus graph, then there exists at least one edge, say uv ∈ E(G), lying on at least two cycles of G. us, the number of cut vertices of G − uv and G is the same, that is, □ A nontrivial connected graph containing no cut vertex is known as a nonseparable graph. A nonseparable subgraph B of a connected graph is said to be maximal nonseparable subgraph if B is not a proper subgraph of any other nonseparable subgraph of G. A block in a graph G is defined as a maximal nonseparable subgraph of G. e next corollary follows directly from Lemma 2.

Corollary 1.
If G is a graph having the minimum (maximum) I f 1 (I f 2 , respectively) value among all graphs of the class V k n , then every block of G is either a cycle or K 2 (complete graph of order 2).

Lemma 3. If G is a graph having the minimum (maximum)
respectively) value among all graphs of the class V k n , then G contains exactly one cycle.
Proof. Suppose to the contrary that G has at least two cycles, say C and C ′ . By Lemma 2, G is a cactus graph.
We claim that each of the cycles C and C ′ has length at least 4. Contrarily, suppose that at least one of C and C ′ has length 3. Without loss of generality, we assume that C has length 3. en, there exists at least one edge, say xy, on C such that G − xy ∈ V k n (if all the three vertices of C are cut vertices, then xy may be chosen arbitrarily; if exactly two vertices of C are cut vertices, then xy may be chosen in such a way that exactly one of x and y is a cut vertex; if exactly one vertex of C is a cut vertex, then xy may be chosen in such a way that neither of x and y is a cut vertex), and hence, by using Lemma 1, we have Let w ∈ V(G) be a cut vertex lying on C such that w 1 and w 2 are the neighbors of w that also lie on C. Let u (different from w) be a neighbor of w 1 . Let xy ∈ E(G) be an edge lying on the cycle C ′ . Let us take G ′ � G − w 1 u, w 2 w, xy + ux, w 2 y . en, we have Since f 1 is strictly increasing and f 2 is strictly decreasing, from equation (9), it follows that 2 Discrete Dynamics in Nature and Society which is a contradiction.

□
A graph containing exactly one cycle is known as a unicyclic graph. Since the cycle graph C n of order n has no cut vertex and it is the only unicyclic graph of minimum degree at least 2, the next result is an immediate consequence of Lemma 3.

Corollary 2.
If G is a graph having the minimum (maximum) I f 1 (, respectively) value among all graphs of the class V k n , then the minimum degree of G is one.

Lemma 4. Let G be a unicyclic graph. Let v ∈ V(G) be a pendent vertex having a neighbor w of degree at least 3 such that w remains a cut vertex in G − v. Let xy ∈ E(G) be an edge lying on the unique cycle of G. It holds that
Proof. We have Due to Lagrange's mean value theorem, there exist numbers θ 1 and θ 2 such that We recall that d G (w) ≥ 3, which implies that θ 1 > θ 2 , and hence, the right-hand side of equation (14) is positive for i � 1 and negative for i � 2 because f 1 is strictly convex and f 2 is strictly concave. □ A path P: v 1 v 2 . . . v r in a graph G is called a pendent path if one of the two vertices v 1 , v r , is pendent and the other has degree greater than 2, and every other vertex (if exists) of P has degree 2. If P: v 1 v 2 . . . v r is a pendent path in which v 1 has degree greater than 2, then v 1 is known as the branching vertex. Two pendent paths are said to be adjacent if they have the same branching vertex.

Lemma 5.
Let G be a unicyclic graph. Let P: wv 1 v 2 . . . v r and P ′ : wu 1 u 2 . . . u q be two adjacent pendent paths in G, where r, q ≥ 2. Let xy ∈ E(G) be an edge lying on the unique cycle of G. It holds that Proof. For simplicity, we take which is positive for i � 1 and negative for i � 2 (see the proof of Lemma 4). where each of the vertices w and w ′ has degree greater than 2.
It holds that Proof. e proof is analogous to that of Lemma 4 and hence omitted.

Main Result
In this section, we state and prove the lower bound on I f 1 and upper bound on I f 2 for the graphs belonging to V k n . e graphs attaining these bounds are characterized as well.
Let C n,k be the graph deduced from the cycle graph C n−k of order n − k and path graph P k+1 of order k + 1 by identifying a vertex of C n−k with an end-vertex of P k+1 (see Figure 1).
where the equality sign in any of these inequalities holds if and only if G is isomorphic to C n,k (see Figure 1).

Proof. Simple calculations yield
We prove the inequality involving I f 1 and the other inequality can be proven in a fully analogous way. Let G min be a graph having the minimum I f 1 value among all graphs of the class V k n . From Lemma 3, it follows that G min contains exactly one cycle. Lemma 4 guaranties that G min does not contain a pendent vertex having a neighbor w of degree greater than 2 such that w remains a cut vertex in G min − v. Also, Lemma 5 forces that G min does not contain any pair of adjacent pendent paths of lengths at least 2. Finally, from Lemma 6, it follows that G min cannot have nonadjacent pendent paths. us, G min is isomorphic to the graph C n,k . □ Discrete Dynamics in Nature and Society Corollary 3. Among all the members of V k n , C n,k is the unique graph attaining the minimum variable sum exdeg index SEI a for a > 1, minimum general zeroth-order Randić index 0 R α for α > 1, minimum multiplicative second Zagreb index Π 2 , and minimum sum lordeg index SL.
Proof. We note that a graph G has the minimum Π 2 value in V k n if and only if G has the minimum ln Π 2 value in V k n . Define g 1 (x) � xa x with a > 1 and x ≥ 1; g 2 (x) � x α with α > 1 and x ≥ 1; g 3 (x) � x ln x with x ≥ 1; and with x ≥ 2. For every i ∈ 1, 2, 3, 4 { }, the function g i is strictly increasing and strictly convex, and hence, from eorem 1, the desired result follows. □ Remark 1. As the extremal graph C n,k mentioned in eorem 1 and Corollary 3 is a molecular graph, C n,k remains extremal if one considers the class of all n-vertex molecular graphs with k ≥ 1 cut vertices and containing at least one cycle instead of V k n in eorem 1 and Corollary 3.

Data Availability
Data about this study may be requested from the authors.

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