Abstract
Let be a finite simple graph with Laplacian polynomial . In an earlier paper, the coefficients and for forests with respect to some degree-based graph invariants were computed. The aim of this paper is to continue this work by giving an exact formula for the coefficient .
1. Definitions and Notations
A simple undirected graph is a pair consisting of a set of vertices and a set of 2-element subsets of . The elements of are called edges, and the number of elements in is called the order of . The notations and denote the number of vertices and edges of , respectively. There are two other graph notations worth mentioning now. The first one is which is the number of edges in with one end point , and the second one is which is defined as the degree of vertex in the line graph of . Obviously, .
We use the notation to denote the path of length two such that vertices and have degree one, and the vertex has degree two. In a similar way, we use the notation to denote a path of length three.
A graph is said to be connected if for arbitrary vertices and in , and there exists a sequence of vertices such that , . The distance between two vertices and in a connected graph , , is defined as the length of a shortest path connecting these vertices and the sum of such numbers is called the Wiener index of , denoted by [1]. The hyper-Wiener index is a generalization of the Wiener index. It was introduced for trees by Randić in 1993 [2] and for a general graph by Klein et al. [3]. This topological index is defined as .
A subgraph of a graph is a graph with vertex set and edge set , such that and . We use the notation to denote that is subgraph of . If , and then the induced subgraph is the graph with vertex set and the edge set , and if , then is a subgraph of , with vertex set and edge set .
The subdivision graph is a graph constructed from by inserting a new vertex on each edge of . It is clear that and .
Suppose is a graph containing two edges and . If the edges and have a common vertex , then we write . In the case that and do not have common vertex, we will say the edges and are independent. If and all pair of edges in are independent, then the set is called a matching for . A -matching is a matching of size , , and the number of such matchings is denoted by . The matching polynomial of and is defined by , where . By definition, , see [4] for more details.
In 1972, Gutman and Trinajsti [5] introduced the first degree-based graph invariant applicable in chemistry. This invariant is the first Zagreb index and can defined by the formula . The second Zagreb index was introduced by Gutman et al. [6] three years later in 1975. The complete history of these graph invariants together with the most important mathematical results about them are reported in [7–10].
The forgotten index of is another variant of the Zagreb group indices defined as [11]. It can be see that , and is the general form of the first Zagreb index. Zhang S. and Zhang H. [12] obtained the extremal values of the general Zagreb index in the class of all unicyclic graphs. Miličević et al. [13] reformulated the first and second Zagreb indices in terms of the edge-degrees instead of the vertex-degrees. These invariants were defined the first and second reformulated Zagreb indices defined as and , respectively.
A -matrix is a matrix whose entries consist only of the numbers and . Suppose is a graph with vertex set . The adjacency matrix of is a -matrix in which if and only if . It is easy to see that is a real symmetric matrix of order and so all of its eigenvalues are real. The matrices and in which and are called the diagonal and Laplacian matrices of , respectively. It is well-known that all eigenvalues of are nonnegative real numbers with as the smallest eigenvalue.
The Laplacian polynomial of a graph is one of the most important polynomial associated to a graph. If is a graph, then the Laplacian polynomial of is the characteristic polynomial of . The roots of this polynomial are called the Laplacian eigenvalues of . Suppose denotes the Laplacian polynomial of . Since the coefficients of the Laplacian polynomial have graph theoretical meaning, some authors took into account the coefficients of this polynomial.
Let be a topological index and be a graph. For simplifying our arguments, we usually write as .
Lemma 1. Suppose is a graph. The following statements hold: (1)(Merris [14] and Mohar [15]) , , and , where is the number of spanning trees of ;(2)(Yan and Yeh [16]) , when is a tree;(3)(Gutman [17]) , when is a tree;(4)(Oliveira et al. [18]) and , where is the number of triangles in .
In [19–21], we proved the following formulas for the coefficients and , when is a forest, respectively:
Suppose and are two arbitrary real numbers. We now define three invariants which is useful in simplifying formulas in our results. These are
Note that the second Zagreb index is just the case of in .
Let and be graphs. Set . If and are two degree-based graph invariants, and then we define two new degree-based topological indices and as and .
Let denote the path graph on vertices. In a recent paper [22], Das et al. presented the following formula for the number of -matchings, , in a graph as
They also proved the following two results:
Lemma 2. Let be a graph with vertices and edges. Then (1).(2).(3).(4),(5).
Lemma 3. Let be a graph with vertices, edges and girth . Then (1).(2).(3).(4).(5).
The following theorem is crucial in our main result [21].
Theorem 4. Let be a graph with edges. Then (1)(2)(3).(4).(5).(6).(7).
2. Laplacian Coefficients and Degree-Based Invariants
The aim of this section is to present an exact formula for the coefficient of the Laplacian polynomial in terms of some degree-based graphs invariants.
Lemma 5. Let be a graph with vertices and edges. Then, .
Proof. Apply definition of , to prove that , and .
The next lemma is a direct consequence of Lemmas 2 and 5 and Theorem 4 (2).
Lemma 6. Let be a graph with vertices and edges. Then, (1).(2).)(3).(4).(5).
It is easy to see that girth . Therefore, Lemma 3 and Theorem 4 (2) imply the following lemma:
Lemma 7. Let be a graph on vertices and edges. Then, (1).(2).(3).(4).(5).
Lemma 8. Let be a graph with vertices and edges. Then, .
Proof. Apply definition of to show that . Now the proof follows from Theorem 4 (2) and simple calculations.
Lemma 9. Let be a graph with vertices and edges. Then, .
Proof. By definition of , . Suppose . Then, . We now replace by to show that . The proof now follows from Theorem 4 (2).
Lemma 10. Let be a graph with vertices and edges. Then, .
Proof. Apply definition of to show that . Now a similar argument as Lemma 9 completes the proof.
Lemma 11. Let be a graph with vertices and edges. Then, .
Proof. Apply definitions of and to write the form . Now a similar argument as Lemma 9 gives the proof.
Lemma 12. Let be a graph with vertices and edges. Then, .
Proof. By definitions of and , we can write Now by simple calculations, we obtain and Theorem 4 (2) gives the result.
Lemma 13. Let be a graph with vertices and edges. Then, .
Proof. By definition of . Now the proof follows from a similar argument as Lemma 9.
Lemma 14. Let be a graph with vertices and edges. Then, .
Proof. By definition of . Now the proof can be completed in a similar way as Lemma 9.
Define nine graph invariants as
Lemma 15. Let be a graph with vertices and edges. Then, .
Proof. By definition of and some tedious calculations, one can see that Therefore, , proving the lemma.
Lemma 16. Let be a graph with vertices and edges. Then, .
Proof. Choose and set and . Apply definition of to prove that . Therefore Now, a similar argument as Lemma 15 gives our result.
Lemma 17. Let be a graph with vertices and edges. Then, .
Proof. Apply definition of and some tedious calculations to show that Now, a similar argument as Lemma 15 gives the proof.
Lemma 18. Let be a graph with vertices and edges. Then, .
Proof. By definition of .
Lemma 19. Let be a graph with vertices and edges. Then, .
Proof. By definition of , as desired.
By a similar arguments as Lemma 15, we can prove the following two lemmas:
Lemma 20. Let be a graph with vertices and edges. Then, .
Lemma 21. Let be a graph with vertices and edges. Then, .
For the sake of completeness, we mention here two results which are useful in our next calculations.
Lemma 22 (see [19]). Let be a graph. Then, .
Theorem 23 (see [4]). Let be a graph with edges and girth . Then, .
Now, Lemmas 2 (1), 3, 15, 16, 17, 18, 19, 20, 21,and 22 and Theorem 23 give the following theorem:
Theorem 24. Let be a graph with edges and girth . Then, .
Lemma 25. Let be a graph with vertices and edges. Then, , , , , , , , , and.
Proof. The proof is straightforward and so it is omitted.
Lemma 26. Let be a graph with vertices and edges. Then, .
Proof. Lemmas 19, 25, and 6 (1) give our result.
Lemma 27. Let be a graph with vertices and edges. Then, (1),(2),(3),(4),(5),(6),(7),(8),(9),(10).
Proof. The proof has tedious calculations similar to Lemma 15.
Lemma 28. Let be a graph with vertices and edges. Then, (1).(2).(3).(4).(5).
Proof. By definitions of , we have proving the lemma.
Theorem 29. Let be a graph with edges. Then, .
Proof. The proof follows from Theorems 24 and 4, Equation (4), and Lemmas 6, 7, 8, 9, 10, 11, 12, 13, 14, 26, 27, and 28.
We mention here a useful result of Zhou and Gutman [23].
Theorem 30. Let be an -vertex forest. Then, , for.
We are now ready to prove the main result of this section.
Theorem 31. Let be a forest with edges. Then, .
Proof. The proof follows from Theorems 29 and 30.
If is an -vertex tree, then . Therefore by previous theorem we have the following corollary:
Corollary 32. Let be a tree on vertices. Then, .
3. Applications
This section aims is to apply our results in Section 3 for computing the Laplacian coefficients , , when is a certain tree. Let be a graph. The number of edges connecting vertices of degree and in a graph is denoted by . Let . As [21], we first assume that be a rooted tree with degree sequence , and is the distance between the center and any pendant vertex, as shown in Figure 1. By definition of , we have , , , , , and . Therefore by Lemma 1, Equations (1) and (2), Corollary 32, and the others simple calculations, we have
For example, see Tables 1 and 2.
Data Availability
All data generated or analyzed during this study are included in this published article. There are no experimental data in this article.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this article.