Ismorphic Lollipop Graphs Are Cospectral with Respect to the Adjacency

Int. J. Contemp. Math. Sciences, Vol. 3, 2008, no. 36, 1783 - 1788

Ismorphic Lollipop Graphs are Cospectral with Respect to the Adjacency

Hong-Qin Bian

College of Information and Electrical Engineering Lanzhou Jiaotong University Lanzhou 730070, P.R. China [email protected]

Qin Li

College of Information and Electrical Engineering Lanzhou Jiaotong University Lanzhou 730070, P.R. China [email protected]

Jing Liu

College of Information and Electrical Engineering Lanzhou Jiaotong University Lanzhou 730070, P.R. China [email protected] Abstract

The lollipop graph,denoted by Ln,p,is obtained by appending a complete graph Kp to a pendant vertex of a path Pn−p.On the basis of discussing the eigenvalues of the lollipop graph,we will show that no two non-isomorphic lollipop graph are cospectral with respect to the adjacency matrix. Mathematics Subject Classiﬁcation: 05C60, 05C90

Keywords: Cospectral graph, Eigenvalues, Lollipop graph

1 Introduction

All graphs considered here are simple and undirected. Let G =(V (G),E(G)) be n vertices and the adjacency matrix A(G).Since matrix A(G) are real 1784 Hong-Qin Bian, Qin Li and Jing Liu

and symmetric ,their eigenvalues are all real numbers. Assume that λ1 ≥ λ2 ≥ ··· ≥ λn are the adjacency eigenvalues of graph G . The adjacency spectrum of the graph G consists of the adjacency eigenvalues (together with their multiplicities), The polynomials PA(G) = |λI − A(G)|, where I is the identity matrix.are deﬁned as the characteristic polynomials of the graph G with respect to the adjacency matrix.Two graphs are said to be cospectral if they have equal spectrum(i.e.,equal characteristic polynomial). In the graph spectrum theorythe problem of spectrum is not just of the important signif- icance in graph theory but also has large application in physics ,chemistry and computer,so more attention is paid to it. In this paper,some more special graphs will be discussed.The lollipop graph,denoted by Ln,p,is obtained by appending a complete graph KP to a pendant vertex of a path Pn−p [1].

2 Preliminary Notes

[2] Lemma 1 Let x1 be a vertex of degree in the graph G and let x2 be the vertex adjacent to x1.Let G1 be the induced subgraph obtained from G by deleting the vertex x1.if x1 and x2 are deleted,the induced subgraph G2 is obtained.Then P λ λP λ − P λ A(G)( )= A(G1)( ) A(G2)( ) [2] Lemma 2 Let Kn,Pn denote the complete graph and the path on n vertices,respectively.Then n−1 PA(Kn)(λ)=(λ − n + 1)(λ +1)

n sin n arccos λ πj (( +1) 2 ) PA(P )(λ)= (λ − 2cos )= n n sin arccos λ j=1 +1 ( 2 ) Let λ =2cosθ, set t1/2 = eiθ,then it is useful to write the adjacency characteristic polynomial of Kn,Pn in the following form: 1/2 −1/2 1/2 −1/2 1/2 −1/2 n−1 PA(Kn)(t + t )=(t + t − n + 1)(t + t +1) 1/2 −1/2 −n/2 n+1 PA(Pn)(t + t )=t (t − 1)/(t − 1) Lemma 3[3] For n × n matrix A and B, the following are equivalent: (i)A and B are cospectral; (ii)A and B have the same characteristic polynomial (iii) tr(Ai)=tr(Bi),for i =1, 2, ···,n.

If A is the adjacency matrix of a graph,then tr(Ai) gives the number of closed walks of length i .So cospectral graphs have the same number of closed walks of a given length i. In particular,they have the same number of edges (take i = 2 ) and triangles (take i = 3). Ismorphic lollipop 1785

3 Main Results

Lemma 3.1 The adjacency characteristic polynomial of the graph Ln,p

In the followingcalculate the adjacency characteristic polynomial of Ln,p . First calculate the adjacency characteristic polynomial of Lp+1,p . Lemma 1 implies that

P λ λP − P λ A(Lp+1,p)( )= A(KP ) A(Kp−1)( ).

a b P λ a λ b − Suppose 1, 1 are the coeﬃcients of A(Lp+1,p)( ) then 1 = ; 1 = 1. second,calculate the adjacency characteristic polynomial of Lp+2,p . Lemma 1 implies that

P λ λP λ − P λ A(Lp+2,p)( )= A(Lp+1,p)( ) A(Kp)( ) λ λP λ − P λ − P λ = ( A(Kp)( ) A(Kp−1)( )) A(Kp)( ) 2 =(λ − 1)PA(Kp))(λ) − λPA(Kp−1)(λ)

Suppose a2,b2 are the coeﬃcients of PA(Pp+2,p)(λ) then a2 = λa1 − 1= 2 λ − 1;b2 = −a1 = −λ. Then,calculate the adjacency characteristic polynomial of Lp+3,p .Lemma 1 implies that,

P λ λP λ − P λ A(Lp+3,p)( )= A(Lp+2,p)( ) A(LP +1,P )( ) 2 = λ((λ − 1)PA(Kp)(λ) − λPA(Kp−1)(λ)) − (λPA(Kp)(λ) − PA(Kp−1)(λ)) 2 2 =(λ(λ − 1) − λ)PA(Kp)(λ) − (λ − 1)PA(Kp−1)(λ)

Suppose a3,b3 are the coeﬃcients of PA(Lp+3,p)(λ) then a3 = λa2 − a1 = 2 2 λ(λ −1)−λ; b3 = −a2 = −(λ −1). Obviously,for n ≥ 3,an = λan−1 −an−2.In 2 the following,solve an. Solve the equation x − λx +1=0. √ x λ+ λ2−4 1 = √2 x λ− λ2−4 . 2 = 2

√ √ a c xn c xn c λ+ λ2−4 n c λ− λ2−4 n c ,c Suppose n = 1 1 + 2 1 = 1( 2 ) + 2( 2 ) , where 1 2 are 2 unknown numbers. Since a1 = λ, a2 = λ − 1, we can obtain the following equations √ √ λ c λ+ λ2−4 c λ− λ2−4 = 1( 2√ )+ 2( 2 √ ) λ2 − c λ− λ2−4 2 c λ− λ2−4 2. 1= 1( 2 ) + 2( 2 )

c 1 √ λ ,c 1 − √ λ Solve these equations, 1 = 2 (1 + λ2−4 ) 1 = 2 (1 λ2−4 ) Then 1786 Hong-Qin Bian, Qin Li and Jing Liu

√ √ 2 2 1 λ λ + λ − 4 n 1 λ λ − λ − 4 n an = (1 + √ )( ) + (1 − √ )( ) λ2 − λ2 − 2 4√ 2 √ 2 √4 2 √ 1 λ + λ2 − 4 λ − λ2 − 4 n λ + λ2 − 4 λ − λ2 − 4 = √ ( − ) ( )i( )n−i λ2 − 4 2 2 2 2 √ √ i=0 n λ + λ2 − 4 λ − λ2 − 4 = ( )i( )n−i i=0 2 2 Let λ =2cosθ,and set t1/2 = eiθ. then n −n/2 n+1 2i−n t (1 − t ) an = t 2 = i=0 1 − t

Then,the adjacency characteristic polynomial of Ln,p is: P t1/2 t−1/2 a P t1/2 t−1/2 − a P t1/2 t−1/2 A(Ln,p)( + )= n−p A(Kp)( + ) n−p−1 A(Kp−1)( + )

1 −n 1/2 p−2 1/2 1/2 = (t 2 (t + t +1) (−p(t + t)+2t +2t + 1)) 1 − t 1 + (t−1 + t−1/2 +1)p−2(p(1 + t1/2) − (t +2t1/2 + 2)) (1 − t) 1 = (ψ(t)+ϕ(t)) 1 − t where −n 1/2 p−2 1/2 1/2 ψ(t)=t 2 (t + t +1) [−p(t + t)+2t +2t +1] ϕ(t)=(t−1 + t−1/2 +1)p−2[p(1 + t1/2) − (t +2t1/2 + 2)] Theorem 3.2 No two non-isomorphic lollipop graphs are cospectral with respect to the adjacency matrix.

Proof. In view of that if two graphs are cospectral they must possess equal number of vertices.Then ,as for two cospectral lollipop graphs,by Lemma 3,they must have the same adjacency characteristic polynomial . Suppose that the graph G1 = Ln,p1 and the graph G2 = Ln,p2 have the same adjacency spectrum,we show that G1 and G2 are isomorphic. By the calculation in section 3,we can easily get the adjacency characteristic polynomials of the graphs G1 and G2 ,respectively. P t1/2 t−1/2 1 ψ t ϕ t A(G1)( + )= 1−t ( 1( )+ 1( )) where −n 1/2 p1−2 1/2 1/2 ψ1(t)=t 2 (t + t +1) [−p1(t + t)+2t +2t +1] n −1 −1/2 p1−2 1/2 1/2 ϕ1(t)=t 2 (t + t +1) [p1(1 + t ) − (t +2t + 2)] Ismorphic lollipop 1787

P t1/2 t−1/2 1 ψ t ϕ t A(G2)( + )= 1−t ( 2( )+ 2( )) where

−n 1/2 p2−2 1/2 1/2 ψ2(t)=t 2 (t + t +1) [−p2(t + t)+2t +2t +1]

n −1 −1/2 p2−2 1/2 1/2 ϕ2(t)=t 2 (t + t +1) [p2(1 + t ) − (t +2t + 2)] Note that P t1/2 t−1/2 P t1/2 t−1/2 A(G1)( + )= A(G2)( + ) Then ψ1(t)+ϕ1(t)) = ψ2(t)+ϕ2(t)) i.e., −n −n 1/2 p1−2 1/2 1/2 1/2 p2−2 1/2 t 2 (t+t +1) [−p1(t +t)+2t+2t +1]−t 2 (t+t +1) [−p2(t +t) n n 1/2 −1 −1/2 p1−2 1/2 1/2 −1 −1/2 +2t+2t +1]+t 2 (t +t +1) [p1(1+t )−(t+2t +2)]−t 2 (t +t p2−2 1/2 1/2 +1) [p2(1 + t ) − (t +2t + 2)] = 0 n Multiply t 2 with both sides of the above equation. 1/2 p1−2 1/2 1/2 1/2 p2−2 1/2 (t + t +1) [−p1(t + t)+2t +2t + 1)] − (t + t +1) [−p2(t + t) 1/2 n −1 −1/2 p1−2 1/2 1/2 n −1 +2t +2t +1]+t (t + t +1) [p1(1 + t ) −(t+2t + 2)] −t (t + −1/2 p2−2 1/2 1/2 t +1) [p2(1 + t ) − (t +2t + 2)] = 0 Then 1/2 p2−2 1/2 1/2 n−p2+2 1/2 p2−2 1/2 (t+t +1) [p2(t +t)−(2t+2t +1)]−t (t+t +1) [p2(1+t ) 1/2 1/2 p1−2 1/2 1/2 n−p1+2 1/2 −(t+2t +2)] = (t+t +1) [p1(t +t)−(2t+2t +1)]−t (t+t p1−2 1/2 1/2 +1) [p1(1 + t ) − (t +2t + 2)] We consider the following cases. Case 1. p ≥ 3 , then p1 ≥ 3p2 ≥ 3, consider the coeﬃcients of one degree terms on both sides of equation,respectively. Then

2 2 p − 3p2 +2 p − 3p1 +2 2 t = 1 t 2 2

p2−3p +2 p2−3p +2 2 2 2 2 p p p p , obviously, 2 = 2 . Clearly 1 = 2 or 1 + 2 =3(a contradic- tion)

Case 2. p<3 then p1 ≤ 2 or p2 ≤ 2.Clearly Ln,p1 Ln,p2 is a path respectively . Obviously if p1 =1then p2 =1;ifp1 =2then p2 =2. In a word p1 = p2 i.e. G1 and G2 are isomorphic.The proof is complete.

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Received: September 24, 2008