On the Spectral Characterization of H-Shape Trees

A graph G is said to be determined by its spectrum if any graph having the same spectrum as G is isomorphic to G. An H-shape is a tree with exactly two of its vertices having maximal degree 3. In this paper, a formula of counting the number of closed 6-walks is given on a graph, and some necessary conditions of a graph Γ cospectral to an H-shape are given.


Introduction
A G is a real symmetric matrix, its eigenvalues must be real, and may be ordered as ( ) ( ) ( ) The sequence of n eigenvalues is called the spectrum of G, the largest eigenvalue ( ) 1 G λ is often called the spectral radius of G.The characteristic polynomial of ( ) A G is called the characteristic polynomial of the graph G and is denoted by ( ) Two graphs are cospectral if they share the same spectrum.A graph G is said to be determined by its spetrum (DS for short) if for any graph H, ( ) ( ) Determining what kinds of graphs are DS is an old problem, yet far from resolved, in the theory of graph spectra.Numerous examples of cospectral but non-isomorphic graphs are reported in literature [1].However, there are few results known about DS graphs.For the background and some recent surveys of the known results about this problem and related topics, we refer the reader to [2]- [6] and references therein.
Because the kind of problems above are generally very hard to deal with, some more modest ones suggested by van Dam and Haemers [2], say, "Which trees are DS?", this problem is also very hard to deal with, because we know a famous result of Schwenk [7], which says that almost all trees have non-isomorphic cospectral mates.
A T-shape ( ) , , T l l l is a tree with exactly one of its vertices having maximal degree 3 such that ( ) More recently, Wang proved that T-shape tree ( ) , , T l l l is DS; Wang and Xu [6] proved that T-shape tree

(
)( ) , , , , 2 2 l l l l l l ≠ − for any positive integer 2 l ≥ .An H-shape is a tree with exactly two of its vertices having maximal degree 3. We denote by

(
)( ) , , , , H l l l l l l l i ≥ ≥ = is an H-shape tree such that ( ) , , , , , , , , , , , H l l l l l u T l l l − = ( ) ( ) , , , , , , , where u and v are the vertices of degree 3.In this paper, we give a formula of counting the number of closed 6-walks on a graph, and give some necessary conditions of a graph Γ cospectral to an H-shape.

Some Lemmas
In the section, we will present some lemmas which are required in the proof of the main result.
Lemma 2.1 [8] The characteristic polynomial of a graph satisfies the following identities: where G e − denotes the graph obtained from G by deleting the edge e and 1 2 G v v − denotes the graph obtained from G by deleting the vertices v 1 , v 2 and the edges incident to it.
Lemma 2.2 [1] Let C n , P n denote the cycle and the path on n vertices respectively.Then  , it is can be write the characteristic polynomial of C n , P n in the following form [6]: where a 0 = 1 and the sum is over all subgraphs γ of G consisting of disjoint edges and cycles, and having i vertices.If γ is such a subgraph then comp(γ) is the number of components in it and cyc(γ) is the number of cycles.Lemma 2.4 [2] [10] Let G be a graph.For the adjacency matrix, the following can be obtained from the spectrum.
1) The number of vertices.
2) The number of edges.
3) Whether G is regular.4) Whether G is regular with any fixed girth.
5) The number of closed walk of any length.6) Whether G is bipartite.

Main Results
The total number of closed k-walks in a graph G, denoted by ( ) k w G .Lemma 3.1 ([6] p. 657) Let G be a graph with e edges, x i vertices of degree i, and y 4-cycles.Then ( ) Lemma 3.2 Let Γ be a graph with n vertices.If Γ cospectral to an H-shape and Γ ≠ W n , then 1) Γ have the same degree sequences as the H-shape tree or Γ have the degree sequences ( ) Proof.Let Γ be a graph with e edges, x i vertices of degree i, and y 4-cycles.By lemma 2.4 we known that cospectral graphs have the same number of edges and closed 4-walks, respectively.Since Γ is cospectral to an H-shape tree, hence by (4) we have from ( 5), we have ( ) the (7) imply to y = 1 or 0. Case 1. y = 1.by (7) we get x 0 = 0 and 3 4 0 . We known that "the spectrum of graph W n is the union of the spectra of the circuit C 4 and the path P n−4 " [1], that is , , , by ( 5) we get 2 6 x n = − and x 1 = 4. Thus Γ have the same degree se- quences as the H-shape tree. If Let G be a graph with e edges, x i vertices of degree i, and z 6-cycles.Then ( ) where p 4 is the number of induced paths of length three and k 1,3 is the number of induced star K 1,3 .
Proof.A close walk of length 6 can be produced from in the following five classes graphs, they are P 2 , P 3 , P 4 , K 1,3 and C 6 .For an edge and a 6-cycle, it is easy to see that the number of close 6-walks equals 2 and 12, respectively.For a P 3 , the number of close 6-walks of a 1-degree vertex is 3 and the number of close 6-walks of the 2-degree vertex is 6, since the number of induced paths of length two is 2 , hence for all induced paths P 3 , the number of close 6-walks is 12 2 . For a P 4 , since the number of close 6-walks of a 1-degree ver-tex is 1 and the number of close 6-walks of a 2-degree vertex is 2, hence for all induced paths P 4 , the number of close 6-walks is 6p 4 .Similarly, for a K 1,3, the number of close 6-walks of a 1-degree vertex is 2 and the number of close 6-walks of the 3-degree vertex is 6, thus for all induced stars K 1,3 , the number of close 6-walks is 12k 1,3 . Corollary 3.4 Let ( ) where ( ) ( ) p + , respectively.The 8(= 4 + 4) is the number of induced paths of through a 3-degree vertex u (or v).If P 4 is such a induced path, then u is an internal vertex in the P 4 and have at least a vertex in the 2) If k = 0, similarly, we have ( ) if we give to a suitable label for the H 1 , by a simple calculation we can get the diagonal matrix of ( ) where ( ) is the number of elements of equals 1 in { } , , , l l l l and p 4 is the number of induced paths of length three and k 1,3 is the number of induced star K 1,3 in Γ.
Proof.If l 1 ≥ 1, by Lemma 3.3 we have ( ) ( ) Similarly, when l 1 = 0 the (10) hold. Definition 1.Let U be a graph obtained from a cycle C g (g is even and 6 ≤ g ≤ n 1 − 2) and a path that identifying an end vertex in the path and any one vertex in the cycle, and uniting an isolated vertex K 1 .
2) There are one 6-cycle in U′ and l 1 = 0, have an element is 1 in { } , , , l l l l .
3) No 6-cycle in U′ and l 1 ≥ 1, have two elements are 1 in { } is even and n 1 + n 2 = n.Let U′ have e edges, x i vertices of degree i, and z 6-cycles.

󲐀
If a graph has the same degree sequences as the H-shape, then Γ is one of the following graphs , Proof.Assume that Γ contains a 1 n P as a connected component, by (11) some li is equal, without loss of generality, let ) ( ) ( ) ( ) ( )  , , , , 1 where 1 2 3 4 5 2 l l l l l n + + + + + =.By (14) we have edge set E. Let ( ) A G be the adjacency matrix of G. Since ( ) vertices, and v is the vertex of degree 3.

If Γ contains a 2 nP
as a connected component, then l 3 = l 5 and l 1 + l 3 + 1 = l 1 , a contradiction. Thus, if a graph ( )n W Γ Γ ≠cospectral to an H-shape and have the same degree sequences as the H-shape, then Γ is one of the following graphs G 3 , G 4 , G 5 (Fig.)uniting some even cycle, respectively, or it is an H-shape.

1 H
t , respectively, we get m 4 = l 4 , m 3 = l 3 and m 2 = l 2 .By + + = + + + + + = , we get m 1 = l 1 , thus 10 Let G 5 be a graph in Figure, then G 5 and H-shape are not cospectral.m m n + + + + + = .Denote the first component by G 5,1 and the second component by G 5,2 Let Γ be a graph with n vertices, e edges, x i vertices of degree i, and z 6-cycles.If Γ cospectral to , without loss of generality, we assume that l 2 ≥ l 3 ≥ l 4 ≥ l 5 and m 2 ≥ m 3 ≥ m 4 ≥ m 5 .Comparing the 4 th lowest term of , we get m 5 = l 5 .Similarly, we comparing the 5 th , 6 th and 7 th lowest term of 1H t . By Lem- ma 2.1 and Lemma 2.3 we have t