Research Article | Open Access
Extremal Matching Energy and the Largest Matching Root of Complete Multipartite Graphs
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 .
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 ). In , Fisher and Ryan deduced some bounds for . Afterwards, many results about the largest matching root have been obtained; see [7–12].
In , 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 . Afterwards, there have been lots of research papers on this topic. A systematic study of this topic can be found in the book . In , Gutman and Wagner pointed out that the matching energy is a quantity of relevance for chemical applications. Moreover, they arrived at the simple relationwhere 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.  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 this section, we first recall an integral formula for computing the value of matching energy which is also given by Gutman and Wagner .
Definition 5 (see ). Let be a simple graph. The matching energy of is
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.
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 .
We first review some useful results of the matching polynomial as the following lemmas.
Lemma 7 (see ). Let be a vertex of . Then
In , Gutman showed some parallel results for the roots of the matching polynomial and the spectra of the characteristic polynomial.
Lemma 8 (see ). 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 .
Definition 9 (see ). Define if for all we have .
We then list some useful propositions about which are proved in .
Proposition 10 (see ). The relation is transitive. And if , then .
Proposition 11 (see ). If is connected and is a proper spanning subgraph of , then .
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 thatBy 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 asLet . For any , we claim thatIn 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 isThus 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 ,