1 Introduction

The Maximal Matching Energy of Tricyclic Graphs

Lin Chen, Yongtang Shi***The corresponding author.

[3mm] Center for Combinatorics and LPMC-TJKLC, Nankai University

Tianjin 300071, P.R. China.

[3mm] E-mail: chenlin1120120012@126.com, shi@nankai.edu.cn

Abstract

Gutman and Wagner proposed the concept of the matching energy (ME) and pointed out that the chemical applications of ME go back to the 1970s. Let be a simple graph of order and be the roots of its matching polynomial. The matching energy of is defined to be the sum of the absolute values of . Gutman and Cvetkoić determined the tricyclic graphs on vertices with maximal number of matchings by a computer search for small values of and by an induction argument for the rest. Based on this result, in this paper, we characterize the graphs with the maximal value of matching energy among all tricyclic graphs, and completely determine the tricyclic graphs with the maximal matching energy. We prove our result by using Coulson-type integral formula of matching energy, which is similar as the method to comparing the energies of two quasi-order incomparable graphs.

1 Introduction

In this paper, all graphs under our consideration are finite, connected, undirected and simple. For more notations and terminologies that will be used in the sequel, we refer to [2]. Let be such a graph, and let and be the number of its vertices and edges, respectively. A matching in a graph is a set of pairwise nonadjacent edges. A matching is called a -matching if the size of is . Denote by the number of -matchings of , where and for or . In addition, define . The matching polynomial of graph is defined as

 α(G)=α(G,x)=∑k≥0(−1)km(G,k)xn−2k.

In 1977, Gutman [4] proposed the concept of graph energy. The energy of is defined as the sum of the absolute values of its eigenvalues, namely,

 E(G)=n∑i=1|λi|,

where denote the eigenvalues of . The theory of graph energy is well developed. The graph energy has been rather widely studied by theoretical chemists and mathematicians. For details, we refer the book on graph energy [24] and reviews [8, 10]. Recently, Gutman and Wagner [13] defined the matching energy of a graph based on the zeros of its matching polynomial [3, 5].

Definition 1.1

Let be a simple graph with order , and be the zeros of its matching polynomial. Then,

 ME(G)=n∑i=1|μi|. (1.1)

Moreover, Gutman and Wagner [13] pointed out that the matching energy is a quantity of relevance for chemical applications. They arrived at the simple relation:

 TRE(G)=E(G)−ME(G),

where TRE() is the so-called “topological resonance energy” of . About the chemical applications of matching energy, for more details see [11, 1, 12].

An important tool of graph energy is the Coulson-type integral formula [4]  (with regard to be a tree ):

 E(T)=2π∫∞01x2ln[∑k≥0m(T,k)x2k]dx, (1.2)

which is valid for any tree (or, more generally, for any forest). Being similar to Eq.(1.2), the matching energy also has a beautiful formula as follows[13].

Proposition 1.2

Let be a simple graph of order , and be the number of its -matchings, . The matching energy of is given by

 ME=ME(G)=2π∫∞01x2ln[∑k≥0m(G,k)x2k]dx. (1.3)

Combining Eq.(1.2) with Eq.(1.3), it immediately follows that: if is a forest, then its matching energy coincides with its energy.

Formula (1.2) implies that the energy of a tree is a monotonically increasing function of any . In particular, if and are two trees for which holds for all , then . If, in addition, for at least one , then . Obviously, by Formula (1.3) and the monotonicity of the logarithm function, the result is also valid for . Thus, we can define a quasi-order” as follows: If two graphs and have the same order and size, then

 (1.4)

And if we say that is -greater than or is -smaller than . If and , the graphs and are said to be -equivalent, denote it by . If , but the graphs and are not -equivalent (i.e., there exists some such that ), then we say that is strictly -greater than , write . If neither nor , the two graphs and are said to be -incomparable and we denote this by .

The relation is an equivalence relation in any set of graphs . The corresponding equivalence classes will be called matching equivalence classes (of the set ). The relation induces a partial ordering of the set . An equivalence class is said to be the greatest if it is greater than any of other class. A class is maximal if there is no other class greater than it. The graphs belonging to greatest (resp. maximal) matching equivalence classes will be said to be -greatest(resp. -maximal) in the set considered.

According to Eq.(1.3) and Eq.(1.4), we have

 G1⪰G2⟹ME(G1)≥ME(G2)

and

 G1≻G2⟹ME(G1)>ME(G2).

It follows that the -greatest graphs must have greatest matching energy, and the -maximal graphs must have greater matching energy than other graphs not to be -maximal.

A connected simple graph with vertices and , , edges are called unicyclic, bicyclic, tricyclic graphs, respectively. Denote by the set of all connected bicyclic graphs of order , and by the set of all connected tricyclic graphs on vertices. Let denote the graph obtained by joining one pendant vertex of to its other two pendant vertices, respectively. Similarly, let be the graph obtained by joining one pendent vertex of to its another three pendent vertices, respectively. Let denote the graph obtained by attaching pendent vertices to one of the four vertices of . Of course, (as shown in Fig. 1.1). Denote by the cycle graph of order and the path graph of order , and let be the graph obtained by connecting two cycles and with a path .

As the research of extremal graph energy is an amusing work (for some newest literatures see [14, 15, 16, 17, 18, 22]), the study on extremal matching energy is also interesting. In [13], the authors gave some elementary results on the matching energy and obtained that for any unicyclic graph , where is the graph obtained by adding a new edge to the star . In [20], Ji et al. proved that for with and , . In [19], the authors characterize the connected graphs (and bipartite graph) of order having minimum matching energy with () () edges. Especially, among all tricyclic graphs of order , , with equality if and only if or . For more results on the matching energy, we refer to [23, 21]. In this paper, we characterize the graphs with the maximal matching energy among all tricyclic graphs, and completely determine the tricyclic graphs with the maximal matching energy.

2 Main Results

In the 1980s, Gutman determined the unicyclic [6], bicyclic [7], tricyclic [9] graphs with maximal matchings, i.e., graphs that are extremal with regard to the quasi-ordering . We introduce the result on tricyclic graphs, which will be used in our proof.

Lemma 2.1 ([9])

In the set of all tricyclic graphs with vertices () the greatest matching equivalence class exists only for and . For there exist two maximal matching equivalence classes. All these equivalence classes possess a unique element, except for , when the number of -greatest graph is two. The corresponding graphs are presented in Fig. 2.2.

Our results are obtained based on the result of Lemma 2.1.

Theorem 2.2

Let with . Then for , ; for , ; for , ; for , ; for , ; for , ; for , ; for , ; for , ; for , , with equality if and only if , where , , , , , , , , , , are the graphs shown in Fig. 2.2.

We will prove our theorem by using Coulson-type integral formula of matching energy Eq.(1.3), which is similar as the method to comparing the energies of two quasi-order incomparable graphs [14, 15, 16, 17, 18, 22]. The following lemmas are needed.

Lemma 2.3 ([25])

For any real number , we have

 X1+X≤ln(1+X)≤X. (2.1)

Let be a simple graph. Let be an edge of connecting the vertices and . By we denote the graph obtained by inserting new vertices (of degree two) on the edge . Hence if has vertices, then has vertices; if , then ; if , then the vertices and are not adjacent in . There is such a result on the number of -matchings of the graph .

Lemma 2.4 ([9])

For all ,

 m(G(e/j+2),k)=m(G(e/j+1),k)+m(G(e/j),k−1).

We will divide Theorem 2.2 into the following two theorems according to the values of .

Theorem 2.5

Let with . Then:
for , ; for , ; for , ; for , ; for , ; for , ; for , ; for , ; for , , where , , , , , , , , , are the graphs shown in Fig. 2.2. In each case, the equality holds if and only if is isomorphic to the corresponding graph with maximal matching energy.

Proof. Let be a graph in with vertices.

For , by Lemma 2.1, is the -greatest graph. We have known that the -greatest graphs must have greatest matching energy. Hence if , then .

When , and are -equivalent, that is, for all . Then by Eq.(1.3), we have . Moreover, if and , then since by Lemma 2.1.

When , both and are -maximal. Thus, if and , then as well as . In addition, we have , , , , , and , , , , , . Make use of Eq.(1.3), by computer-aided calculations, we get and . Therefore, .

For , both and are -maximal. Similarly, by the help of computer, we get , , , , , , respectively. Therefore, if , then we have .

The proof of the theorem is complete.

Theorem 2.6

Let with . Then , with equality if and only if , where is the graph shown in Fig. 2.2.

Proof. By Lemma 2.1, both and are -maximal. The -maximal graphs must have greater matching energy than other graphs not to be -maximal. Thus, if and , then and . It is sufficient to show that . We will make full use of the definition of matching polynomial and Eq.(1.3).

Assume that , then and . According to Lemma 2.4, we have

 α(G(e/j+2),x)=∑k≥0(−1)km(G(e/j+2),k)xn−2k=∑k≥0(−1)km(G(e/j+1),k)xn−2k+∑k≥0(−1)km(G(e/j),k−1)xn−2k=x∑k≥0(−1)km(G(e/j+1),k)x(n−1)−2k−∑k≥0(−1)k−1m(G(e/j),k−1)x(n−2)−2(k−1)=xα(G(e/j+1),x)−α(G(e/j),x).

By the definition of , clearly, and , where and are the graphs shown in Fig. 2.3.

Therefore, both and satisfy the recursive formula

 f(n,x)=xf(n−1,x)−f(n−2,x).

The general solution of this linear homogeneous recurrence relation is

 f(n,x)=C1(x)(Y1(x))n+C2(x)(Y2(x))n,

where , . By some elementary calculations, we can easily obtain the values of as follows.

In the following, we first consider . It is easy to calculate the number of -matchings of and : , , , , for ; , , , , , for . Then by Lemma 2.4, we can calculate the values of for all and . Thus, take the initial values as:

 α(G(1)(e/4),x)=x11−13x9+59x7−114x5+89x3−21x=C1(x)(Y1(x))11+C2(x)(Y2(x))11;α(G(1)(e/5),x)=x12−14x10+71x8−162x6+165x4−63x2+5=C1(x)(Y1(x))12+C2(x)(Y2(x))12.

It is easy to check that and . Therefore, by solving the two equalities above, we get

 C1(x)=Y1(x)α(G(1)(e/5),x)−α(G(1)(e/4),x)(Y1(x))13−(Y1(x))11

and

 C2(x)=Y2(x)α(G(1)(e/5),x)−α(G(1)(e/4),x)(Y2(x))13−(Y2(x))11.

Define

 A1(x)=Y1(x)α(G(1)(e/5),x)−α(G(1)(e/4),x)(Y1(x))13−(Y1(x))11.
 A2(x)=Y2(x)α(G(1)(e/5),x)−α(G(1)(e/4),x)(Y2(x))13−(Y2(x))11.

Then we have .

Now we consider . Similarly, we get: , , , , , , for ; , , , , , , , for . Then calculate the values of for all and by using Lemma 2.4. We can then take the initial values as:

 α(G(2),x)=x11−13x9+59x7−112x5+84x3−20x=C1(x)(Y1(x))11+C2(x)(Y2(x))11;α(G(2)(e/1),x)=x12−14x10+71x8−161x6+164x4−68x2+8=C1(x)(Y1(x))12+C2(x)(Y2(x))12.

Therefore, we obtain that:

 C1(x)=Y1(x)α(G(2)(e/1),x)−α(G(2),x)(Y1(x))13−(Y1(x))11

and

 C2(x)=Y2(x)α(G(2)(e/1),x)−α(G(2),x)(Y2(x))13−(Y2(x))11.

Define

 B1(x)=Y1(x)α(G(2)(e/1),x)−α(G(2),x)(Y1(x))13−(Y1(x))11.
 B2(x)=Y2(x)α(G(2)(e/1),x)−α(G(2),x)(Y2(x))13−(Y2(x))11.

Then we have

From the expression of , we have

 α(G,ix)=∑k≥0(−1)km(G,k)(ix)n−2k=in∑k≥0m(G,k)xn−2k=(ix)n∑k≥0m(G,k)x−2k,

where . Thus, by Eq.(1.3), we get

 ME(Gn(1))−ME(Gn(2))=2π∫∞01x2ln[∑k≥0m(Gn(1),k)x2k]dx−2π∫∞01x2ln[∑k≥0m(Gn(2),k)x2k]dx=2π∫∞01x2ln∑k≥0m(Gn(1),k)x2k∑k≥0m(Gn(2),k)x2kdx=2π∫∞0ln∑k≥0m(Gn(1),k)x−2k∑k≥0m(Gn(2),k)x−2kdx=2π∫∞0lnα(Gn(1),ix)α(Gn(2),ix)dx=2π∫∞0lnA1(ix)(Y1(ix))n+A2(ix)(Y2(ix))nB1(ix)(Y1(ix))n+B2(ix)(Y2(ix))ndx. (2.2)

By the definition of and , we have and . Now we define , , and

 f1=iα(G(1)(e/4),ix)=x11+13x9+59x7+114x5+89x3+21x
 f2=α(G(1)(e/5),ix)=x12+14x10+71x8+162x6+165x4+63x2+5
 g1=iα(G(2),ix)=x11+13x9+59x7+112x5+84x3+20x
 g2=α(G(2)(e/1),ix)=x12+14x10+71x8+161x6+164x4+68x2+8.

Then we have and . Moreover, It follows that

 A1(ix)=iZ1(x)f2+if1(iZ1(x))13−(iZ1(x))11=Z1(x)f2+f1(Z1(x))11((Z1(x))2+1)
 A2(ix)=iZ2(x)f2+if1(iZ2(x))13−(iZ2(x))11=Z2(x)f2+f1(Z2(x))11((Z2(x))2+1)
 B1(ix)=iZ1(x)g2+ig1(iZ1(x))13−(iZ1(x))11=Z1(x)g2+g1(Z1(x))11((Z1(x))2+1)
 B2(ix)=iZ2(x)g2+ig1(iZ2(x))13−(iZ2(x))11=Z2(x)g2+g1(Z2(x))11((Z2(x))2+1).

Note that , , and . We will distinguish with two cases.

Case 1. is odd.

Now we have

 lnA1(ix)(Y1(ix))n+2+A2(ix)(Y2(ix))n+2B1(ix)(Y1(ix))n+2+B2(ix)(Y2(ix))n+2−lnA1(ix)(Y1(ix))n+A2(ix)(Y2(ix))nB1(ix)(Y1(ix))n+B2(ix)(Y2(ix))n=ln(1+K0(x)H0(n,x)),

where

 K0(x)=(A1(ix)B2(ix)−A2(ix)B1(ix))((Y1(ix))2−(Y2(ix))2)=(f2g1−f1g2)x=−x18−19x16−146x14−588x12−1342x10−1750x8−1253x6−460x4−68