Báo cáo toán học: "The orderings of bicyclic graphs and connected graphs by algebraic connectivity" ppsx

The orderings of bicyclic graphs and connected graphsby algebraic connectivity ∗ Jianxi Li Department of Mathematics & Information Science Zhangzhou Normal University Zhangzhou, Fujian,

The orderings of bicyclic graphs and connected graphs

by algebraic connectivity ∗

Jianxi Li

Department of Mathematics & Information Science

Zhangzhou Normal University Zhangzhou, Fujian, P R China

fzjxli@tom.com

Ji-Ming Guo

Department of Applied Mathematics China University of Petroleum Dongying, Shandong, P R China jimingguo@hotmail.com

Wai Chee Shiu

Department of Mathematics Hong Kong Baptist University Kowloon Tong, Hong Kong, P R China

wcshiu@hkbu.edu.hk Submitted: May 31, 2010; Accepted: Nov 15, 2010; Published: Dec 3, 2010

Mathematics Subject Classifications: 05C50 Keywords: bicyclic graph, connected graph, algebraic connectivity, order

Abstract The algebraic connectivity of a graph G is the second smallest eigenvalue of its Laplacian matrix Let Bnbe the set of all bicyclic graphs of order n In this paper,

we determine the last four bicyclic graphs (according to their smallest algebraic connectivities) among all graphs in Bn when n > 13 This result, together with our previous results on trees and unicyclic graphs, can be used to further determine the last sixteen graphs among all connected graphs of order n This extends the results of Shao et al [The ordering of trees and connected graphs by their algebraic connectivity, Linear Algebra Appl 428 (2008) 1421-1438]

∗ Supported by the National Science Foundation of China (No.10871204); the Fundamental Research Funds for the Central Universities (No.09CX04003A); FRG, Hong Kong Baptist University.

1 Introduction

v in G and d(v) = |N(v)| be the degree of v For any e ∈ E(G), we use G − e to denote

Let A(G) and D(G) be the adjacency matrix and the diagonal matrix of vertex degrees

of G, respectively The Laplacian matrix of G is defined as L(G) = D(G)−A(G) It is easy

to see that L(G) is a symmetric positive semidefinite matrix having 0 as an eigenvalue

µ1(G) > µ2(G) > · · · > µn(G) = 0,

popularly known as the algebraic connectivity of G and is usually denoted by α(G)

It has be found a lot of applications in theoretical chemistry, control theory, combinatorial

n

graph spectra One of which is how to order graphs according to their (Laplacian) eigen-values Hence ordering graphs with various properties by their spectra, specially by their

deter-mined the last four trees (according to their smallest algebraic connectivities) among all

can be combined into the following theorem

Kirk-land [5]

(2) α(Cn,g+1) > α(Cn,g)

3

T

5

T

4

T

6

T

7

deter-mined the graphs with the second and the third smallest algebraic connectivities among

following theorem

α(U) > α(U7) Moreover, α(U1) < α(U2) < α(U3) < α(U4) < α(U5) < α(U6) < α(U7)

1

U

2

4

6

in Bn In this paper, we further extend their result to the last four bicyclic graphs These together with the previous result on the trees and unicyclic graphs, we can extend the ordering of connected graphs by their smallest algebraic connectivities form the last six connected graphs to the last sixteen connected graphs

2 Preliminaries

In this section, we present some lemmas which will be used in the subsequent sections

graphs with girth 3 or 4

n \ {U1, U2, U3, U5, U6, U7} with n > 13, where Ui are shown

α(U7)

n \ {Cn,4 ∼= U4} with n > 8, α(U) > α(U7).

3 Bicyclic graphs

Firstly, we introduce some notations that are used in this section Let ∞ (coalescence

B = θ4(T1, T2, T3, T4) We also write B = θ4(i, j, k, l) instead of θ4(Pi+1, Pj+1, Pk+1, Pl+1),

Now, we give the first four bicyclic graphs of order n > 13 with smallest algebraic connectivity

v

5

v

3

v

2

v

1

v

4

v

3

v

2

1

v

q

¥

Figure 3: Bicyclic graphs ∞ and θ

More-over, α(B1) < α(B2) < α(B3) < α(B4), where B1, B2, B3, B4 are shown in Fig 4

1

3

B

4

B

1

u

2

u

1

v

2

v

1

u

2

u

1

u

2

4

u

3

u

4

u

α(B3) < α(B4)

α(B) > α(B − e) > α(Cn,l) > α(Cn,5) > α(U7) = α(B4)

In the following we suppose that l 6 4 We consider the following two cases

(a) l = 4

(b) k = l = 3

some edge, say e, in Ck or Cl such that B − e ∈ Un3 and B − e does not

imply that

must be a bicyclic graph which consists of the graph H (here H is a bicyclic

1

2

u

2

v

{ P uv

Figure 5: Bicyclic graph H, where u 6= v

n

α(B) > α(B − u1u2)(or α(B − v1v2)) > α(U7) = α(B4)

U1, U2, U3, U5, U6, U7 Thus the result follows

α(B†) > α(B†− u1u2)(or α(B†− v1v2)) > α(U7) = α(B4)

(a) l = 4

edges, say e, in E(Ck) ∩ E(Cl) such that B − e ∈ Un4 and B − e does not

n or U6

n,

(b) k = l = 3

n, B can be rewrote as B = θ4(T1, T2, T3, T4), and |V (Ti)| = ni

n and

that

n1 >2 or n2 >2 Therefore, B ∼= θ4(T1, P1, P1, P1) or B ∼= θ4(P1, T2, P1, P1)

If θ4(T1, P1, P1, P1) does not isomorphic to B3 and θ4(P1, T2, P1, P1) does not isomorphic to B2 That is, θ4(T1, P1, P1, P1)−v1v3 and θ4(P1, T2, P1, P1)−v1v3

imply that

α(θ4(T1, P1, P1, P1)) > α(θ4(T1, P1, P1, P1) − v1v3) > α(U7) = α(B4) and

α(θ4(P1, T2, P1, P1)) > α(θ4(P1, T2, P1, P1) − v1v3) > α(U7) = α(B4)

4 Connected graphs

In Section 3, we determined the last four bicyclic graphs according to their smallest

the orderings of the trees and unicyclic graphs, in this section, we extend the ordering of connected graphs from the last six connected graphs to the last sixteen connected graphs Before giving the main result of this section, the following preliminary results are needed Lemma 4.1 Let G be a connected graph of order n > 13 which contains exactly n + 2

Proof Let v be a vertex of degree 3 in G, e be an edge on some cycle C of G such that

Lemma 4.2 Let G be a connected graph of order n > 13 with maximum degree ∆(G) = 4

with n > 13, we consider the following three cases

Case 1 m = n − 1, n, n + 1

Case 2 m = n + 2

Let v be a vertex of degree 4 in G, e be an edge on some cycle C of G such that

If G′ ∼= B3 (shown in Fig. 4), since G does not isomorphic to G14, then in

Case 3 m > n + 3

In this case, it suffices to prove that for any connected graph G of order n > 13

if G with m > n + 3, we may delete m − (n + 3) edges from G such that the resulting graph (with n + 3 edges) is connected) Let v be a vertex of degree 4

in G, e be an edge on some cycle C of G such that e is not incident with v, and

Now, we give the main result of this section

G1, , G16 are shown in Fig 7 and G1 ∼= T1, G2 ∼= T2, G3 ∼= U1, G4 ∼= T3, G5 ∼= U2, G6 ∼

B1, G7 ∼= T4, G8 ∼= U3, G9 ∼= U4, G10∼= B2, G11 ∼= T5, G12 ∼= U5, G13 ∼= B3, G15 ∼= T6, G16 ∼

U6

α(G14) < α(G15) = α(G16) and α(B4) > α(G16)

Case 1 ∆(G) = 2

2(n−1)

2

G

3

G

7

G

8

G

4

6

G

9

G

11

G

12

G

13

G

14

G

10

G

15

we have

α(Cn) > α(Pn−1) > α(T7) > α(G15) = α(G16)

Case 2 ∆(G) = 3

that the resulting graph is also a connected graph with n + 2 edges Thus the

Case 3 ∆(G) = 4

Case 4 ∆(G) > 5

Then G contains a spanning tree T with ∆(T ) = ∆(G) > 5 Clearly, T does

The authors are indebted to the anonymous referees for their valuable comments and suggestions

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