The maximum Mostar indices of unicyclic graphs with given diameter
Introduction
Let be a simple and undirected graph. Define being the order of . Denote by (or for short) the distance between two vertices in , which is defined as the edge number of the shortest path between them. For each edge , letand . Thenis called the contribution of . The Mostar index of is defined as the sum of all edge contributions. That is,
The Mostar index was proposed in [1], which measures peripherality in chemistry and complex networks, and measures how far a graph is from distance-balanced [2], [3], [4], [5], [6], where a distance-balanced graph is the graph with the Mostar index being zero. With respect to the index, the extremal graphs among trees [1], [7], [8], tree-like chemical graphs [9], [10], [11], [12], [13], [14], unicyclic graphs [1], bicyclic graphs [15], cacti [16] and so on, were studied. In [8], the Mostar index was found to be a good predictor for the total surface area of octane isomers, by being compared with other distance-based graph invariants. A recent article [17] gathered many known extremal results concerning the index, and stated some modifications of the index, such as the edge Mostar index [18], [19], [20], [21].
A unicyclic graph is the graph with exactly one cycle. The diameter of a graph is the greatest distance between two of its vertices. Let be the graph set with order and diameter For a graph set , let be the graph set with the greatest Mostar index. The aim of the paper is to determine and their Mostar indices.
A cycle (resp. a path) with edges is called a -cycle (resp. a -path) for short. Let , , , and be the graphs obtained by the following steps, respectively.
- (1)
(resp. ): Let be a 3-cycle (resp. a 4-cycle) and be a vertex in . First, attach (resp. ) pendent edges at . Second, attach a pendent -path and a pendent -path at . See Fig. 1.
- (2)
: Let be a 3-cycle and be an edge in . First, attach pendent edges at . Second, if is odd, then either attach a pendent -path at and a pendent -path at , or attach a pendent -path at each of and ; see Fig. 2(a). If is even, then attach a pendent -path at and a pendent -path at ; see Fig. 2(b).
- (3)
: Let be a 4-cycle and be an edge in . First, attach pendent edges at . Second, if is odd, then attach a pendent -path at and a pendent -path at ; see Fig. 3(a). If is even, then either attach a pendent -path at and a pendent -path at , or attach a pendent -path at and a pendent -path at ; see Fig. 3(b).
- (4)
: Let be a 4-cycle and be an edge in . First, attach a pendent -path at . Second, if , then attach pendent edges at ; see Fig. 4(a). If and is even, then attach pendent edges at in where or ; see Fig. 4(b). If and is odd, then attach pendent edges at in where ; see Fig. 4(c).
Theorem 1.1
Suppose and . Let be a unicyclic graph with order and diameter .
- (1)
If and , thenwith the equality if and only if when ; and or when .
- (2)
If and , or , thenwith the equality only if or when is even and ; and otherwise.
- (3)
Let If , then with the equality if and only if . If , thenwith the equality if and only if or when ; and when .
Theorem 1.1 is proved in Section 3. The proof idea is as follows: first, show ; second, determine by straightforward calculations. The calculations are based on basic cut method, which is stated in Section 2. General cut method was already used in counting Mostar index [22] recently and in counting the closely related Szeged index [23] before. For more about cut method, one can also be referred to the latest survey paper [24]. The other method we use in the proof is doing some graph transformations, such as the moving operation which is also stated in Section 2.
Section snippets
Cut method and graph transformations
For subgraphs and in a graph , denote by (or ) the least distance between one vertex in and the other vertex in . For , letbe the contribution of .
Let be a connected unicyclic graph with being the unique cycle. For a vertex in , denote by the component of which contains , where and are the two edges incident to in .
Let be a cut edge in , then denote by (resp. ) the component of which contains
Proof of Theorem 1.1
First, we have the following two lemmas. Let . Lemma 3.1 Suppose and . Then . Proof Let be in where and , with being a path of length and being the unique cycle of . Suppose for some . Let . Note that . Let . Without loss of generality, suppose when , and when . Claim 3.1 Each vertex in has degree 2. Proof
Concluding remarks and a further research problem
In this paper, the unicyclic graphs with order and diameter , having the greatest Mostar index are determined. A bicyclic graph is a connected graph whose edge number equals to the order plus one. The following is a natural sequential research problem. Problem 1 Among all bicyclic graphs with order and diameter , identify the graphs having the largest and smallest Mostar indices.
Declaration Of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgments
Kecai Deng is partially supported by Fundamental Research Funds for the Central Universities of Huaqiao University (No. ZQN-904), and Fujian Province University Key Laboratory of Computational Science, School of Mathematical Sciences, Huaqiao University, Quanzhou, China.
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