Abstract
A bond incident degree (BID) index of a graph is defined as , where denotes the degree of a vertex of , is the edge set of , and is a real-valued symmetric function. The choice in the aforementioned formula gives the variable sum exdeg index , where is any positive real number. A cut vertex of a graph is a vertex whose removal results in a graph with more components than has. A graph of maximum degree at most 4 is known as a molecular graph. Denote by the class of all -vertex graphs with cut vertices and containing at least one cycle. Recently, Du and Sun [AIMS Mathematics, vol. 6, pp. 607–622, 2021] characterized the graphs having the maximum value of from the set for . In the present paper, we not only characterize the graphs with the minimum value of from the set for , 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 -vertex molecular graphs with cut vertices and containing at least one cycle.
1. Introduction
Graph invariants of the following form are known as the bond incident degree (BID) indices [1–4]:where denotes the degree of a vertex of the graph , is the edge set of , and is a real-valued symmetric function. In this paper, we are concerned with the following type [5] of the BID indices:where , is a strictly increasing and strictly convex function, while is a strictly decreasing and strictly concave function.
If is a positive real number, then the variable sum exdeg index of a graph can be defined as
The graph invariant 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 [7–12].
A cut vertex of a graph is a vertex whose removal results in a graph with more components than has. An -vertex graph is a graph of order . Let be the set of all -vertex graphs with 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 from the set when . The 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 from the set for , but also we solve a more general problem concerning the BID indices and . By using the obtained general result, we also characterize the graphs with the minimum general zeroth-order Randić index (see [14]) when , minimum multiplicative second Zagreb index (see [15]), and minimum sum lordeg index (see [16–18]) from the class , where
We note that equation (5) giveswhich is minimum in a given class of graphs if and only if 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 -vertex molecular graphs with cut vertices and containing at least one cycle.
All the graphs considered in this paper are connected. The notation and terminology that are used in this paper but not defined here can be found in some standard graph-theoretical books, like [19, 20].
2. Lemmas
A cut vertex of is a vertex whose removal increases the number of components of . Denote by the class of all connected -vertex graphs with 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 and are adjacent vertices of a graph , then it holds that
Proof. The proof follows directly from the definitions of and .
A cactus graph is a connected graph in which every pair of cycles has at most one vertex in common.
Lemma 2. If is a graph having the minimum (maximum) (, respectively) value among all graphs of the class , then is a cactus graph.
Proof. If is not a cactus graph, then there exists at least one edge, say , lying on at least two cycles of . Thus, the number of cut vertices of and is the same, that is, . But Lemma 1 forces that and , which is a contradiction.
A nontrivial connected graph containing no cut vertex is known as a nonseparable graph. A nonseparable subgraph of a connected graph is said to be maximal nonseparable subgraph if is not a proper subgraph of any other nonseparable subgraph of . A block in a graph is defined as a maximal nonseparable subgraph of . The next corollary follows directly from Lemma 2.
Corollary 1. If is a graph having the minimum (maximum) (, respectively) value among all graphs of the class , then every block of is either a cycle or (complete graph of order 2).
Lemma 3. If is a graph having the minimum (maximum) (, respectively) value among all graphs of the class , then contains exactly one cycle.
Proof. Suppose to the contrary that has at least two cycles, say and . By Lemma 2, is a cactus graph.
We claim that each of the cycles and has length at least 4. Contrarily, suppose that at least one of and has length 3. Without loss of generality, we assume that has length 3. Then, there exists at least one edge, say , on such that (if all the three vertices of are cut vertices, then may be chosen arbitrarily; if exactly two vertices of are cut vertices, then may be chosen in such a way that exactly one of and is a cut vertex; if exactly one vertex of is a cut vertex, then may be chosen in such a way that neither of and is a cut vertex), and hence, by using Lemma 1, we have and , a contradiction.
Let be a cut vertex lying on such that and are the neighbors of that also lie on . Let (different from ) be a neighbor of . Let be an edge lying on the cycle . Let us take . Then, we haveSince is strictly increasing and is strictly decreasing, from equation (9), it follows thatwhich is a contradiction.
A graph containing exactly one cycle is known as a unicyclic graph. Since the cycle graph of order 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 is a graph having the minimum (maximum) (, respectively) value among all graphs of the class , then the minimum degree of is one.
Lemma 4. Let be a unicyclic graph. Let be a pendent vertex having a neighbor of degree at least 3 such that remains a cut vertex in . Let be an edge lying on the unique cycle of . It holds that
Proof. We haveDue to Lagrange’s mean value theorem, there exist numbers and such thatWe recall that , which implies that , and hence, the right-hand side of equation (14) is positive for and negative for because is strictly convex and is strictly concave.
A path in a graph is called a pendent path if one of the two vertices , , is pendent and the other has degree greater than 2, and every other vertex (if exists) of has degree 2. If is a pendent path in which has degree greater than 2, then is known as the branching vertex. Two pendent paths are said to be adjacent if they have the same branching vertex.
Lemma 5. Let be a unicyclic graph. Let and be two adjacent pendent paths in , where . Let be an edge lying on the unique cycle of . It holds that
Proof. For simplicity, we take . Now, we havewhich is positive for and negative for (see the proof of Lemma 4).
Lemma 6. Let be a unicyclic graph. Let and be two nonadjacent pendent paths in , where each of the vertices and has degree greater than 2. It holds that
Proof. The proof is analogous to that of Lemma 4 and hence omitted.
3. Main Result
In this section, we state and prove the lower bound on and upper bound on for the graphs belonging to . The graphs attaining these bounds are characterized as well.
Let be the graph deduced from the cycle graph of order and path graph of order by identifying a vertex of with an end-vertex of (see Figure 1).
Theorem 1. If , thenwhere the equality sign in any of these inequalities holds if and only if is isomorphic to (see Figure 1).
Proof. Simple calculations yieldWe prove the inequality involving and the other inequality can be proven in a fully analogous way. Let be a graph having the minimum value among all graphs of the class . From Lemma 3, it follows that contains exactly one cycle. Lemma 4 guaranties that does not contain a pendent vertex having a neighbor of degree greater than 2 such that remains a cut vertex in . Also, Lemma 5 forces that does not contain any pair of adjacent pendent paths of lengths at least 2. Finally, from Lemma 6, it follows that cannot have nonadjacent pendent paths. Thus, is isomorphic to the graph .
Corollary 3. Among all the members of , is the unique graph attaining the minimum variable sum exdeg index for , minimum general zeroth-order Randić index for , minimum multiplicative second Zagreb index , and minimum sum lordeg index .
Proof. We note that a graph has the minimum value in if and only if has the minimum value in . Define with and ; with and ; with ; and with . For every , the function is strictly increasing and strictly convex, and hence, from Theorem 1, the desired result follows.
Remark 1. As the extremal graph mentioned in Theorem 1 and Corollary 3 is a molecular graph, remains extremal if one considers the class of all -vertex molecular graphs with cut vertices and containing at least one cycle instead of in Theorem 1 and Corollary 3.
Data Availability
Data about this study may be requested from the authors.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This research has been funded by Scientific Research Deanship at University of Ha’il, Saudi Arabia, through project number BA-2032.