Matching Energy of Unicyclic and Bicyclic Graphs with a Given Diameter

# Matching Energy of Unicyclic and Bicyclic Graphs with a Given Diameter

###### Abstract

Gutman and Wagner proposed the concept of 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 . In this paper, we characterize the graphs with minimal matching energy among all unicyclic and bicyclic graphs with a given diameter .

Lin Chen, Jinfeng Liu, Yongtang Shi111The corresponding author.

Center for Combinatorics and LPMC-TJKLC, Nankai University, Tianjin 300071, P.R. China

E-mails: chenlin1120120012@126.com, ljinfeng709@163.com, shi@nankai.edu.cn

Key words: graph; matching energy; energy; diameter

## 1 Introduction

In this paper, all graphs under our consideration are finite, connected, undirected and simple. For more notations and terminology that will be used in the sequel, we refer to [2]. Let be a simple undirected graph with order and be the adjacency matrix of . The characteristic polynomial of , denoted by , is defined as

 ϕ(G)=det(xI−A(G))=n∑i=0ai(G)xn−i,

where is the identity matrix of order . The roots of the equation , denoted by , are the eigenvalues of . The energy of , denoted by , is defined as the sum of the absolute values of the eigenvalues of , that is,

 E(G)=n∑i=1|λi|.

The concept of the energy of simple undirected graphs was introduced by Gutman in [15] and now is well-studied. For more results about graph energy, we refer the readers to recent papers [8, 9, 12, 32, 34], two surveys [16, 17] and the book [31]. There are various generalizations of graph energy, such as Randić energy [3, 10], Laplacian energy [7], distance energy [36], incidence energy [4, 5], energy of matrices [14] and energy of a polynomial [33], etc.

Let be a simple graph with vertices and edges. Denote by the number of -matchings( the number of selections of independent edges the number of -element independent edge sets) of . Specifically, and for or . It is both consistent and convenient to define . The matching polynomial of the graph is defined as

 α(G)=α(G,μ)=∑k≥0(−1)kmk(G)μn−2k. (1)

Recently, Gutman and Wagner [23] defined the matching energy of a graph based on the zeros of its matching polynomial [13, 21].

###### 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|. (2)

Moreover, Gutman and Wagner [23] 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 [18, 1, 20].

For the coefficients of , let . Note that , , and is the number of edges of . For convenience, let if . In [19, 24], we have

 (3)

Thus is a monotonically increasing function of .

Being similar to Eq.(3), the matching energy also has a beautiful formula as follows[23]. Eq.(4) could be considered as the definition of matching energy, in which case Eq.(2) would become a theorem.

###### Theorem 1.1

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≥0mk(G)x2k]dx. (4)

By Eq.(4) and the monotony of the function logarithm, we can define a quasi-order” as follows: If two graphs and have the same order and size, then

 G1⪰G2⟺mk(G1)≥mk(G2)% for all k.

If and there exists some such that , then we write . Clearly,

Notice that when , we may not deduce that . However, if is any simple connected graph with vertices other than , where is a star of order , then not only [23] but also . Based on the quasi-order, there are some more extremal results on matching energy of graphs [6, 26, 27, 30].

In this paper, we characterize the graphs with minimal matching energy among all unicyclic and bicyclic graphs with a given diameter .

## 2 Preliminaries

The following result gives two fundamental identities for the number of -matchings of a graph (see [13, 21]).

###### Lemma 2.1

Let be a simple graph, be an edge of , and be the set of all neighbors of in . Then we have

 mk(G)=mk(G−uv)+mk−1(G−u−v), (5)
 mk(G)=mk(G−u)+j∑i=1mk−1(G−u−vi). (6)

From Lemma 2.1, we know that . And we can also obtain that

###### Lemma 2.2

Let be a simple graph and be a subgraph(resp. proper subgraph) of . Then (resp. ).

A connected graph with vertices and edges is called a unicyclic graph. Obviously, a unicyclic graph has exactly one cycle. A connected graph with vertices and edges is called a bicyclic graph. Let be the class of connected unicyclic graphs with vertices, be the class of unicyclic graphs with vertices and diameter , where . Let be the class of bicyclic graphs with vertices and be the class of bicyclic graphs in with diameter , where . Let be the path with vertices and be the complete graph with vertices.

When , , is the unique graph in . When , , contains no graphs. When , , has two graphs and (see Figure 1). Clearly, , i.e., is the unique graph with minimal matching energy in . When , the graph obtained by attaching pendant vertices to a vertex of a triangle is the unique graph in . Thus, we just consider the case in which . In section of our paper, we will prove that for , the graph is the unique graph in with minimal matching energy, where the graph is shown in Figure 2.

When , , has two graphs and (see Figure 3). By Lemma 2.1 and simple calculation, we can get , hence is the unique graph in with minimal matching energy. Therefore, we only consider the case in which . In section , we will show that is the unique graph with minimal matching energy for , where the graph is shown in Figure 2. Furthermore, we also pay our attention to the case .

Let be the class of trees with vertices and diameter , where . If , then . For , let denote the graph obtained by attaching pendent vertices to an end vertex of . Specially, . Obviously, is the unique tree in and is the unique tree in .

Let be two graphs with vertices. Now we introduce a quasi-order defined in [31]: If for all , then we write . If and there exists an such that , then we write . The following lemmas are relevant results on this quasi-order.

###### Lemma 2.3 ([22, 40])

For and ,

 Pn≻1Pi∪Pn−i≻1P1∪Pn−1.

For , .

Let and . Then .

If , then .

###### Lemma 2.7 ([39])

For , we have , where if , and if and .

If is an acyclic graph, then [22] and for all . Thus, the quasi-order (resp. ) in Lemmas 2.32.7 can be replaced by (resp. ), and the results also work.

By Lemma 2.1 and the definition of the quasi-order , it is easy to see that the following lemma holds.

###### Lemma 2.8

Let , and (resp. ) be a pendant edge with the pendant vertex (resp. ) of the graph (resp. ). If and , or and , then .

The following lemmas will be needed in our paper, which are obtained based on the previous results.

###### Lemma 2.9

For , .

Proof. Since is a proper subgraph of , then by Lemma 2.2, we can get . Similarly, we also have .

###### Lemma 2.10

For , .

Proof. By Lemmas 2.1, 2.2 and 2.6,

 mk(Un,d) ≥ mk(Un−1,d−1)+mk−1(Tn−2,d−2) ≥ mk(Un−1,d−1)+mk−1(Td−1,d−3) = mk(Un,d−1).

Furthermore, . It follows that . Therefore, .

Similarly, we have

For , .

## 3 Unicyclic graphs with a given diameter

Now we consider the minimal matching energy of graphs in with . We first discuss the case .

###### Lemma 3.1

Let with and . Then .

Proof. We will prove the lemma by induction on .

If , then is isomorphic to one of the following graphs (see Figure 4).

It is easy to get

 α(U18,6)=μ8−8μ6+19μ4−13μ2+1;   α(U28,6)=μ8−8μ6+18μ4−11μ2+1;
 α(U38,6)=μ8−8μ6+18μ4−12μ2+1;   α(U48,6)=μ8−8μ6+19μ4−14μ2+2;
 α(U58,6)=μ8−8μ6+18μ4−12μ2+2;   α(U8,6)=μ8−8μ6+18μ4−11μ2.

And then it is obvious that for , i.e., is the unique graph in with minimal matching energy.

If , then is isomorphic to one of the following graphs (see Figure 5).

We can obtain that

 α(U19,7)=μ9−9μ7+26μ5−26μ3+6μ;   α(U29,7)=μ9−9μ7+25μ5−23μ3+5μ;
 α(U39,7)=μ9−9μ7+25μ5−24μ3+6μ;   α(U49,7)=μ9−9μ7+25μ5−24μ3+5μ;
 α(U59,7)=μ9−9μ7+26μ5−27μ3+8μ;   α(U69,7)=μ9−9μ7+25μ5−24μ3+6μ;

along with

 α(U9,7)=μ9−9μ7+25μ5−23μ3+4μ.

It now immediately follows that for , i.e., is the unique graph in with minimal matching energy.

Now suppose that the result holds for graphs in and . Let and , where .

Let (resp. ) be a pendant vertex, adjacent to (resp. ), which has the largest distance to a vertex on the unique cycle of (resp. ). Then the degree of is 2. So is . Hence , and , .

Since , we have either or . By the induction hypothesis, we have and , or and . By Lemma 2.8, .

###### Theorem 3.2

Let with , and . Then .

Proof. We prove the result by induction on .

When , by Lemma 3.1, we have . Let and suppose that the result holds for . Now suppose that . Let be the vertex of degree 3 in and be a vertex on the quadrangle that is adjacent to . By Lemma 2.1,

 mk(Un,d) = mk(Un,d−u′v′)+mk−1(Un,d−u′−v′) = mk(Tn,d)+mk−1(Pd−3∪Sn−d+1).

For , . By Lemma 2.4, . And by Lemmas 2.3 and 2.4, . Thus . Therefore, we may suppose that the unique cycle of is with . Let be a diametrical path of . Then one of and must be a pendant vertex.

Case 1 All pendant vertices are on .

Since , then . Thus there are at least two adjacency vertices, say and , on which lie outside such that , and , where . By Lemmas 2.5 and 2.6, , . We also have by Lemmas 2.2 and 2.4. Hence, .

Note that , so for all . Moreover, since , there exists some such that , i.e., . Thus .

Case 2 There is at least one pendant vertex outside .

Let be a pendant vertex of adjacent to the vertex of degree . Then , and

Subcase 2.1 There is a pendant vertex outside such that its neighbor lies on .

Since outside , then . Consequently, by the induction hypothesis, .

If lies outside , then . Thus .

Suppose that lies on , then and have common vertices, say with .

If , i.e., is the unique common vertex of and , then . Since

 mk(Pi∪Pd−i∪P2)≥mk(Pd)≥mk(Td,d−2)

and , then . Therefore, .

If . For , . So . Otherwise, for or , say . Then , where is obtained by attaching a pendant vertex to vertex of the path . For ,

 mk(Pi∪T1)≥mk(Pi∪Td−i+1,d−i−1)≥mk(Td,d−2).

If , then , hence . Otherwise, is a proper subgraph of , then . Thus we always have . Therefore, .

We have proved that . Then by Lemma 2.8, we obtain .

Subcase 2.2 The neighbor of any pendant vertex outside also lies outside .

If there is a pendant vertex such that its neighbor lies outside , then or , where with .

If every pendant vertex outside is adjacent to a vertex on , then we choose a pendant vertex , adjacent to such that or , where with , and .

Hence there are three possibilities: , or .

First, suppose that , then

 mk(Cr∪Pj∪Pd−j)≥mk(P3∪Pj∪Pd−j)≥mk(Td,d−2).

In particular, . Thus, .

Next, suppose that , then .

Finally, suppose that For with . By the induction hypothesis, . Thus

 G−u−v⪰Pj∪G′′⪰Pj∪Us′,d′⪰Pj∪Ud−j+1,d−j−1.

For ,

 mk(Pj∪Ud−j+1,d−j−1)≥mk(Pj∪Td−j+1,d−j−1)≥mk(Td,d−2).

Furthermore, . It follows that

According to the arguments above, we have proved that . On the other hand, . Thus by Lemma 2.8, .

Combining Cases 1 and 2, we conclude that also holds for with and , which yields the result.

## 4 Bicyclic graphs with a given diameter

In what follows we state some new definitions and notations. For a graph , it has either two or three distinct cycles. If has exactly two cycles, suppose that the lengths of them are and respectively. If has three cycles, then any two cycles must have at least one edge in common, and we may choose two cycles of lengths of and with common edges such that and . Then, in any case, we choose two cycles and in . For convenience, let and . If and have no common edges, then and are connected by a unique path , say from to . Let be the length of . If and have exactly common edges, and thus have exactly common vertices, say, , then is the third cycle of , where . If we write , then . Denote by the diameter of .

Now we turn our attention to the minimal matching energy of graphs in with . We first deal with the case .

###### Lemma 4.1

Let with , and . Then .

Proof. By induction on to prove this fact.

For and , there are only finitely many graphs we need to consider. Then by Lemma 2.1 and direct check, we can get .

Suppose that the result holds for all graphs in and , where . Let and .

Case 1 There is a pendent vertex in such that the degree of its neighbor is .

In this case, and . Since , then or . By the induction hypothesis, and , or and . Hence, .

Case 2 The neighbor of any pendent vertex has degree at least or there is no pendent vertex.

Then is isomorphic to some , (see Figure 6), or contains one triangle or one quadrangle which has at most one common vertex with the other cycle that is a triangle or a quadrangle.

If is isomorphic to , then by Lemmas 2.1, 2.5, 2.6 and Theorem 3.2,

 mk(G) = mk(G−u1v2)+mk−1(G−u1−v2) ≥ mk(Un,n−3)+mk−1(Tn−2,d−1) ≥ mk(Un,n−3)+mk−1(Pn−6∪S4) = mk(Bn,n−3).

Moreover, , thus .

If is isomorphic to , then by Lemmas 2.1, 2.3 and Theorem 3.2,

 mk(G) = mk(G−u1u0)+mk−1(G−u1−u0) ≥ mk(Un,n−3)+mk−1(P2∪Tn−4,n−6) ≥ mk(Un,n−3)+mk−1(P2∪Pn−5)+mk−2(P2∪Pn−7) ≥ mk(Un,n−3)+mk−1(P3∪Pn−6)+mk−2(Pn−6) = mk(Bn,n−3).

Similarly, , thus .

Otherwise, contains one triangle or one quadrangle which has at most one common vertex with the other cycle that is a triangle or a quadrangle. Choose and as above. Let .

If , then is isomorphic to , or in Figure 7, where the black vertices may not occur. Similarly, we can obtain that