Complexity
Volume 2019 (2019), Article ID 9728976, 7 pages
https://doi.org/10.1155/2019/9728976

Research Article

Extremal Matching Energy and the Largest Matching Root of Complete Multipartite Graphs

Xiaolin Chen and Huishu Lian

Department of Mathematics, China University of Mining and Technology, Xuzhou 221116, China

Correspondence should be addressed to Huishu Lian; lhs6803@126.com

Received 15 February 2019; Accepted 4 April 2019; Published 16 April 2019

Academic Editor: Dimitri Volchenkov

Copyright © 2019 Xiaolin Chen and Huishu Lian. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

The matching energy of a graph was introduced by Gutman and Wagner, which is defined as the sum of the absolute values of the roots of the matching polynomial . The largest matching root is the largest root of the matching polynomial . Let denote the complete -partite graph with order , where . In this paper, we prove that, for the given values and , both the matching energy and the largest matching root of complete -partite graphs are minimal for complete split graph and are maximal for Turán graph .

1. Introduction

Let be a finite simple graph of order with vertex set and edge set . A matching of is a set of independent edges in , and a -matching of is a matching of that has exactly edges. By we denote the number of -matchings in . It is easy to verify that, for and , . And when , is the size of . For convenience, we define . The matching polynomial [1–3] of a graph is defined as

The matching polynomial has been widely studied and has many important applications in statistical physics and chemistry; see [1–4]. The largest root of the matching polynomial , denoted by , is called the largest matching root of . The matching root has been much studied in algebraic combinatorics (see the book [5]). In [6], Fisher and Ryan deduced some bounds for . Afterwards, many results about the largest matching root have been obtained; see [7–12].

In [13], Gutman and Wagner introduced the matching energy (ME) of a graph , which is defined as the sum of the absolute values of the roots of the matching polynomial of . Note that the concept of the energy of a simple undirected graph was introduced by Gutman in [14]. Afterwards, there have been lots of research papers on this topic. A systematic study of this topic can be found in the book [15]. In [13], Gutman and Wagner pointed out that the matching energy is a quantity of relevance for chemical applications. Moreover, they arrived at the simple relation

where is the so-called “topological resonance energy” of the graph . For more information about the applications of the matching energy, we refer the readers to [16, 17]. Recently, there have been some results on the matching energy of graphs; see [18–31]. In spite of this, few results are known for the matching energy of complete multipartite graphs. Note that Stevanović et al. [32] studied the energy of complete multipartite graphs, which inspires us to study the matching energy of complete multipartite graphs.

For a fixed value , let denote the complete -partite graph with order . There are two important complete -partite graphs which have many implications in graph theory. One is complete split graph, denoted by , which is defined as . The other is Turán graph , which is defined as the complete -partite graph satisfying for any two distinct numbers .

In this paper, we show that, among all the complete -partite graphs with given order , the graph with minimal matching energy is the complete split graph and the graph with maximal matching energy is Turán graph . Moreover, we prove that, among all the complete -partite graphs with given order , the graph with minimal is the complete split graph and the graph with maximal is Turán graph . In fact, we deduce the following theorems.

Theorem 1. For the fixed values and , and equality holds if and only if .

Theorem 2. For the fixed values and , and equality holds if and only if .

Theorem 3. For the fixed values and , and equality holds if and only if .

Theorem 4. For the fixed values and , and equality holds if and only if .

In Section 2, we prove Theorems 1 and 2. In Section 3, we give the proofs of Theorems 3 and 4.

2. Proofs of Theorems 1 and 2

In this section, we first recall an integral formula for computing the value of matching energy which is also given by Gutman and Wagner [13].

Definition 5 (see [13]). Let be a simple graph. The matching energy of is

By this definition, we can deduce that if for every , then . Owing to the above observation, we derive the following lemma which is critical to the proofs of Theorems 1 and 2.

Lemma 6. For a complete -partite graph with , we have , and , for .

Proof. Let denote the graph . Let be the partition classes with for and . Without loss of generality, we just prove the case .
For , . For , we will prove this lemma by induction on the number . For the case , we first delete the vertex and the edges incident to it from , and then we get a new graph which is isomorphic to . Next, we add a new vertex into the graph and add some edges such that we get a graph which is isomorphic to . In other words, these operations can be thought as the vertex being moved from the second partition class to the first partition class. Thus, for convenience, we denote the new vertex by . Now, we compare the number of -matchings between and . Choosing a -matching in , if the edge lies in the matching , is not a -matching of . Otherwise, the matching is a -matching of . For a -matching in , when there exists an edge between and , is not in . Otherwise, is a -matching of . It follows that . Since and , we get .
Given , suppose that the lemma is right for , that is, for . For , just like the above operations, we move the vertex from into , then we get a new graph which is isomorphic to . For a -matching in , when there exists an edge between and in the matching , is not in . Otherwise, the matching is a -matching of . For a -matching in , when there exists an edge between and , is not in . Otherwise, is a -matching of . Then . Since , , then . Thus, we complete the proof.

From this lemma, we give the proofs of Theorems 1 and 2.

Proof of Theorem 1. Suppose that is the complete multipartite graph with the smallest matching energy. If there are two parameters and satisfying , then . From Lemma 6, we get , and for It implies that . It is a contradiction. Hence, satisfies that all parameters equal except one parameter equal to . Thus, .

Proof of Theorem 2. Suppose that is the complete multipartite graph with the largest matching energy. If there are two parameters and satisfying , from Lemma 6, we get , and for . Owing to Definition 5, we can deduce that . It is a contradiction. Hence, for any , we have . It means that each parameter equals either or , i.e., is isomorphic to .

3. Proofs of Theorems 3 and 4

We first review some useful results of the matching polynomial as the following lemmas.

Lemma 7 (see [5]). Let be a vertex of . Then

In [33], Gutman showed some parallel results for the roots of the matching polynomial and the spectra of the characteristic polynomial.

Lemma 8 (see [33]). If is a subgraph of , then . If is connected and is a proper subgraph of , then .

Next, we recall a definition which is introduced in [34].

Definition 9 (see [34]). Define if for all we have .

We then list some useful propositions about which are proved in [34].

Proposition 10 (see [34]). The relation is transitive. And if , then .

Proposition 11 (see [34]). If is connected and is a proper spanning subgraph of , then .

Now, we are ready to give the following lemma which is the key to the proofs of Theorems 3 and 4.

Lemma 12. For a complete -partite graph with , we have .

Without loss of the generality, we just need to prove the following lemma.

Lemma 13. For a complete -partite graph with , we have .

Proof. Using Lemma 7, we get the following two equations:

By subtraction, we deduce that

By Definition 9, it suffices to show that, for every , . In the sequence, we will study the structure of the polynomial (6) and deduce an important recurrent formula.
By Lemma 7, we first give two different expansions of .

Then, combining with (7) and (8), we derive that

Suppose , we get the following recurrent formula of the right side of (6):

Next, we will repeatedly use the above recurrent formula (10). For convenience, we introduce a new notation . Then, (10) can be rewritten as

Let . For any , we claim that

In the above formula, the symbol , where are two nonnegative integers, is called the falling factorial which is defined as , and it means the number of -permutations of in combinatorics. The symbol is defined as . Note that , , if . The numbers in the above formula are the multinomial coefficients and . In the reminder of paper, for the brevity of deduction, we give the following notes. If is a negative integer, then . If there exists a number for , then .
Now, we prove Equation (12) by induction.
When , it is (11). Suppose the above equation is true for . We now consider the case .
For , by (11), we can get its expanded formula.

Then we can obtain that

By expanding the last formula, we get that

We observe that the general term in the last summation is , in which are nonnegative integers and . We then study the coefficient of .
By analyzing the above formula, we find that the coefficient of consists of parts. For every , the th part comes from the expansion of , and it is

Thus the coefficient of is

It is well known that

It follows the coefficient of is equal to

Owing to the above analysis, we conclude that

Hence, (12) holds for . By induction, (12) is true for any .
When , we get the final expanded form.

Note that . Now, considering the element , for , we have

From the above calculations, for , we can deduce the following result:

Thus,

For a set of nonnegative integers , is a proper subgraph of . By Lemma 8, we get . When , we have . By (24), we have . It follows that .
We thus complete the proof.

Now, it is time to prove Theorems 3 and 4.

Proof of Theorem 3. Suppose that the largest matching root of complete -partite graphs is minimal for . If there are two parameters and satisfying , then . From Lemma 12, we get . By Proposition 10, we get . It is a contradiction. Hence, satisfies that all parameters are equal to except one parameter equal to . Thus, .

Proof of Theorem 4. Suppose that the largest matching root of complete -partite graphs is maximal for . If there are two parameters and satisfying , then . Owing to Lemma 12, we have . By Proposition 10, it implies that . It is a contradiction. Hence, for any , we have . It means that each parameter equals either or , i.e., is isomorphic to .

4. Conclusion

Recently, there are many results about the matching energy and matching roots of graphs [7, 11, 12, 18–23, 25–28, 31, 34], due to their important applications in statistical physics and chemistry. In this paper, we investigate the matching energy and the largest matching root among all the complete -partite graphs with given order , respectively. We find that their minimal graphs are both complete split graph , and the maximal graphs are both Turán graph . In fact, complete split graph and Turán graph are two important graph classes which have extensive applications in extremal graph theory and network. For more details, we refer to [35]. In addition, there are many other chemical invariants, such as Laplacian energy [36], Randić energy [37], and incidence energy [38]. In the future, we would like to characterize the extremal graphs of the above invariants among the complete -partite graphs.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

This work was supported by National Natural Science Foundation of China (no. 11601511), the Natural Science Foundation of Jiangsu Province (nos. BK20160237 and BK20150169).

References

E. J. Farrell, “An introduction to matching polynomials,” Journal of Combinatorial Theory, Series B, vol. 27, no. 1, pp. 75–86, 1979.

C. D. Godsil and I. Gutman, “On the theory of the matching polynomial,” Journal of Graph Theory, vol. 5, no. 2, pp. 137–144, 1981.

I. Gutman, “The matching polynomial,” MATCH Communications in Mathematical and in Computer Chemistry, vol. 6, pp. 75–91, 1979.

O. J. Heilmann and E. H. Lieb, “Theory of monomer-dimer systems,” Communications in Mathematical Physics, vol. 25, pp. 190–232, 1972.

C. Godsil, Algebraic Combinatorics, Chapman and Hall, London, UK, 1993.

D. C. Fisher and J. Ryan, “Bounds on the largest root of the matching polynomial,” Discrete Mathematics, vol. 110, no. 1-3, pp. 275–278, 1992.

S. Akbari, P. Csikvári, A. Ghafari, S. Khalashi Ghezelahmad, and M. Nahvi, “Graphs with integer matching polynomial zeros,” Discrete Applied Mathematics: The Journal of Combinatorial Algorithms, Informatics and Computational Sciences, vol. 224, pp. 1–8, 2017.

E. Ghorbani, “Graphs with few matching roots,” Graphs and Combinatorics, vol. 29, no. 5, pp. 1377–1389, 2013.

I. Gutman, “A survey on the matching polynomial,” in Graph Polynomials, M. Dehmer, Y. Shi, I. Gutman, and X. Li, Eds., pp. 73–95, CRC Press, Boca Raton, FL, USA, 2016.

W. Liu, Q. Guo, Y. Zhang, L. Feng, and I. Gutman, “Further results on the largest matching root of unicyclic graphs,” Discrete Applied Mathematics: The Journal of Combinatorial Algorithms, Informatics and Computational Sciences, vol. 221, pp. 82–88, 2017.

H. Zhang, R. Lin, and G. Yu, “The largest matching root of unicyclic graphs,” Information Processing Letters, vol. 113, no. 19-21, pp. 804–806, 2013.

H. Zhang, G. Yu, and S. Li, “Graphs with six distinct matching roots,” Information Processing Letters, vol. 115, no. 5, pp. 521–526, 2015.

I. Gutman and S. Wagner, “The matching energy of a graph,” Discrete Applied Mathematics, vol. 160, no. 15, pp. 2177–2187, 2012.

I. Gutman, “The energy of a graph,” Berichte der Mathematisch-Statistischen Sektion im Forschungszentrum Graz, vol. 103, pp. 1–22, 1978.

X. Li, Y. Shi, and I. Gutman, Graph Energy, Springer, New York, NY, USA, 2012.

I. Gutman, “Acyclic systems with extremal Hückel π-electron energy,” Theoretical Chemistry Accounts, vol. 45, no. 2, pp. 79–87, 1977.

I. Gutman, M. Milun, and N. Trinajstić, “Graph theory and molecular orbitals. 19. Nonparametric resonance energies of arbitrary conjugated systems,” Journal of the American Chemical Society, vol. 99, no. 6, pp. 1692–1704, 1977.

L. Chen and J. Liu, “The bipartite unicyclic graphs with the first largest matching energies,” Applied Mathematics and Computation, vol. 268, pp. 644–656, 2015.

L. Chen and J. Liu, “Extremal values of matching energies of one class of graphs,” Applied Mathematics and Computation, vol. 273, pp. 976–992, 2016.

L. Chen, J. Liu, and Y. Shi, “Matching energy of unicyclic and bicyclic graphs with a given diameter,” Complexity, vol. 21, no. 2, pp. 224–238, 2015.

L. Chen, J. Liu, and Y. Shi, “Bounds on the matching energy of unicyclic odd-cycle graphs,” MATCH - Communications in Mathematical and in Computer Chemistry, vol. 75, no. 2, pp. 315–330, 2016.

L. Chen and Y. Shi, “The maximal matching energy of tricyclic graphs,” MATCH Communications in Mathematical and in Computer Chemistry, vol. 73, no. 1, pp. 105–119, 2015.

X. Chen, X. Li, and H. Lian, “The matching energy of random graphs,” Discrete Applied Mathematics, vol. 193, pp. 102–109, 2015.

I. Gutman, “Matching energy,” in Graph Energies-Theory and Applications, I. Gutman and X. Li, Eds., pp. 167–190, University of Kragujevac, Kragujevac, Serbia, 2016.

S. Ji, X. Li, and Y. Shi, “Extremal matching energy of bicyclic graphs,” MATCH Communications in Mathematical and in Computer Chemistry, vol. 70, no. 2, pp. 697–706, 2013.

S. Ji and H. Ma, “The extremal matching energy of graphs,” Ars Combinatoria, vol. 115, pp. 343–355, 2014.

H.-H. Li, Y.-X. Zhou, and L. Su, “Graphs with extremal matching energies and prescribed parameters,” MATCH - Communications in Mathematical and in Computer Chemistry, vol. 72, no. 1, pp. 239–248, 2014.

S. Li and W. Yan, “The matching energy of graphs with given parameters,” Discrete Applied Mathematics, vol. 162, pp. 415–420, 2014.

J. Monsalve, J. Rada, and Y. Shi, “Extremal values of energy over oriented bicyclic graphs,” Applied Mathematics and Computation, vol. 342, pp. 26–34, 2019.

K. Xu, K. C. Das, and Z. Zheng, “The minimal matching energy of -graphs with a given matching number,” MATCH - Communications in Mathematical and in Computer Chemistry, vol. 73, no. 1, pp. 93–104, 2015.

K. Xu, Z. Zheng, and K. Das, “Extremal t-apex trees with respect to matching energy,” Complexity, vol. 21, no. 5, pp. 238–247, 2016.

D. Stevanovic', I. Gutman, and M. U. Rehman, “On spectral radius and energy of complete multipartite graphs,” Ars Mathematica Contemporanea, vol. 9, no. 1, pp. 109–113, 2015.

I. Gutman, “A note on analogies between the characteristic and the matching polynomial of a graph,” Publications de l’institut Mathématique, vol. 31, no. 45, pp. 27–31, 1982.

X. Chen, X. Li, and H. Lian, “Solution to a conjecture on the maximum skew-spectral radius of odd-cycle graphs,” The Electronic Journal of Combinatorics, vol. 22, no. 1, pp. P1–71, 2015.

A. Branstadt, V. B. Le, and J. P. Spinrad, Graph Classes: A Survey, SIAM Monographs on Discrete Mathematics and Applications, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, USA, 1999.

I. Gutman and B. Zhou, “Laplacian energy of a graph,” Linear Algebra and its Applications, vol. 414, no. 1, pp. 29–37, 2006.

I. Gutman, B. Furtula, and S. Bozkurt, “On Randic' energy,” Linear Algebra and its Applications, vol. 442, no. 1, pp. 50–57, 2014.

S. B. Bozkurt and D. Bozkurt, “On incidence energy,” MATCH - Communications in Mathematical and in Computer Chemistry, vol. 72, no. 1, pp. 215–225, 2014.