_{1}

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The nullity of a graph is the multiplicity of the eigenvalue zero in its spectrum. In this paper we show the expression of the nullity and nullity set of unicyclic graphs with *n* vertices and girth *r*, and characterize the unicyclic graphs with extremal nullity.

Let _{1} and G_{2} is denoted by_{n}, P_{n}, C_{n} and K_{n}, respectively. An isolated vertex is sometimes denoted by K_{1}.

Let A(G) be the adjacency matrix of G. The eigenvalues

called its nullity and is denoted by η(G). Let r(G) be the rank of A(G). Clearly,

A graph is said to be singular (nonsingular) if its adjacency matrix A(G) is a singular (nonsingular) matrix.

In [

G (corresponding to an alternant hydrocarbon), if

represents is unstable. The nullity of a graph is also important in mathematics, since it is related to the singularity of A(G). The problem has not yet been solved completely. Some results on trees and it’s line graphs, bipartite graphs, unicyclic graphs, bicyclic graphs and tricyclic graphs are known (see [

A unicyclic graph is a simple connected graph in which the number of edges equals the number of vertices.

The length of the shortest cycle in a graph G is called the girth of G, denoted by g(G). If G is a unicyclic graph, then the girth of G is the length of the only cycle in G.

Let U_{n} be the set of all unicyclic graph with n vertices and let U(n, r) be the set of all unicyclic graphs with n vertices and girth r. A subset N of {0, 1, 2, ..., n} is said to be the nullity set of U(n, r) provided that for any kÎN, there exists at least one graph

A matching of G is a set of independent edges of G, a maximal matching is a matching with maximum possible number of edges. The collection of all maximal matching is denoted by M(G), for any

It is difficult to give an expression of the nullity of a graph, so many papers give that the upper bound of the nullity of some specific graphs and characterized the extremal graphs attaining the upper bound (see [

Theorem 1.1 [

Theorem 1.1 implies to

In this paper we show the expression of the nullity and nullity set of unicyclic graphs with n vertices and girth r, and characterize the unicyclic graphs with extremal nullity. For terminology and notation not defined here we refer to [

The following lemmas are needed, Lemmas 2.1 and Lemma 2.3 are clear.

Lemma 2.1 Let H be an induced subgraph of G. Then

Lemma 2.2 Let H be an induced subgraph of G. Then

Proof.

Lemma 2.3 Let_{ }then

where

Lemma 2.4 [

Let

Lemma 2.5

So we discuss that r < n in the following unicyclics. Let U_{0}(n, r) be the set of all unicyclic graphs with n vertices and girth r and r < n, let U_{0,1}(n, r) be the subset of U_{0}(n, r) with odd girth r and let U_{0,2}(n, r) be the subset

of U_{0}(n, r) with even girth r, clearly

Lemma 2.6 [

The characteristic polynomial of graph G is denoted by

Lemma 2.7 [^{n-i} is

where the sum is over all subgraphs H of G consisting of disjoint edges and cycles, and having i vertices. If H is such a subgraph then k(H) is the number of components in it and c(H) is the number of cycles.

Let

edges and cycles.

In [

Theorem 3.1 For a graph G,

1)

2)

where the sum is over all spanning subgraphs H of G consisting of disjoint edges and cycles.

Proof. By (1) it is clear.

By (1) we know also that

Corollary 3.1 For a graph G,

Let U be a unicyclic graph with girth r, Let H be a subgraphs of U consisting of disjoint edges and cycles with maximum possible number of vertices. Let H be the collection of all H. Since U is unicyclic graph, then H have

two types: _{2}, where C_{r} is induced subgraph of U and mP_{2} is disjoint union of

m edges P_{2}. Let

Since U doesn’t contains a subgraph G_{1} consisting of disjoint edges and cycles, such that

Corollary 3.2 Let U be a unicyclic graph with girth r, then

Theorem 3.2 Let

1) there exist_{r});

2) for any_{r}).

Where C_{r} is induced subgraph of U.

Proof. Let _{r} be an induced subgraph of U. By Corollary 2.2, we only need to discuss

that _{r}, in nature order.

Case 1.

is even,

then

Case 2.

Subcase 2.1 There exist

_{0}, H_{1} and

H_{2} are same, and we call H_{1} and H_{2} are conjugate subgraph of H_{0}. Since r/2 is odd, hence for any HÎH, the

number of component of H have the same odevity, hence

Subcase 2.2 There doesn’t exist

hence

Case 3. _{r}).

Subcase 3.1 There exist H_{0}ÎH_{1}, where

we have

Since we know that there exist_{r}),

hence we assume that _{3}, such that they not all belong to

E(C_{r}). Except H_{3}, if there exist others _{i} (i ≥ 4), such that they not all belong to E(C_{r}), then we have

and

Subcase 3.2 There aren’t exist

Case 4. r ≡ 0(mod 4) and for any_{r}).

In this case, for any

is independent edges in C_{r.} For the same _{1}, let

is also independent edges in C_{r}, then

exist a conjugate graph

that is

ponding two conjugate subgraphs

where

H of U consisting of disjoint edges and cycles, and having m(U) ? 1 edges. Clearly there exist a

(m(U) ? 1)-matching, such that there exist r/2 − 1 edges belong in E(C_{r}) and

Let C_{r} be a cycle and let P_{n}_{−}_{r} be a path. Suppose that v is a vertex of C_{r} and u is a pendant vertex of P_{n}_{−}_{r}. Joining v and u by an edge, the resulting graph (

Corollary 3.3 Let

Proof. Since

Case 1.

have

Case 2.

dant edge belong to M, that is for any r/2 edges in M, it not all belong to E(C_{r}), so

Let

Corollary 3.4 [

Corollary 3.5 Let

Proof. Let

“⇒” If

Case 1. If n is even, then

Case 2. If n is odd, then

“⇐”

Case 1. If n is even and U contains PM, then

2.2, η(U) = 0.

Case 2. If n is odd and

Corollary 3.6 Let

_{r}).

Proof. Let

“⇒” If

is even, hence

PM, such that they not all belong to E(C_{r}). Otherwise, by Theorem 2.2 we have

“⇐”

Case 1. If

Case 2. If _{r}), then by Theorem 2.2 we have

An edge belonging to a matching of a graph G is said to cover its two end-vertices. A vertex v is said to be perfectly covered (PC) if it is covered in all maximal matching of G [

Any vertex adjacent to a pendent vertex is a PC-vertex. However, there may be exist PC-vertices adjacent to no pendent vertex. For instance, the central vertex in the path on an odd number of vertices is PC.

Let _{r}. Let _{r}, by adding r_{i}

_{r}, respectively. Where

vertex joint to the PC-vertex, but the sum of number of all pendant vertices is n ? r. For r = 5 and 6, an

Let

Clearly

C_{r} is also the PC-vertices of U, where C_{r} is inducted subgraph of U.

Let d(v, G) denote the distance from a vertex v to the graph G, if vÎV(G), then

Corollary 3.7 Let

Proof. Since

“⇒” Let

pendant v of U,

vertex v in U, such that

So for any pendant vertex of U,_{r}, if there exist pendant

edges for every vertices of C_{r} in U, then

tion. Hence there exist pendant edges for part of vertices of C_{r} in U. If there exist (r+1)/2 + 1 vertices in C_{r} such

that every vertex have pendant edges, then

tradiction. So there exist at most (r+1)/2 vertices, such that every vertex have pendant edges, that is all pendant

vertices of U joint to at most (r+1)/2 vertices in C_{r}. In the neighbor vertices of all pendant vertices of U, if there

exist (r?1)/2 PC-vertices and one non PC-vertex of C_{r}, then

Thus all pendant vertices of U are joint to the PC-vertices of C_{r}, thus

“⇐” Let

Let u be a vertex of C_{r}, and let v be a k-degree vertex of K_{1,k+1}. Joining u and v by a path P_{l}, the resulting graph is denoted by U(r, l, k + 1), where

For convenience, we call the star in U(r, 2, k + 1) is pendant star. Let U′(r, l, k) be a unicyclic graph come from U(r, l, k + 1), by removing a pendant edge and adding it to another vertex of C_{r}, where r + l + k = n (See

Corollary 3.8 Let

and

Proof. Since

“⇒” Let

Case 1.

respectively. Then

thus

Case 2.

Subcase 2.1. There exist vÎU, such that

graph of U, then there exist

M, it not all belong to E(C_{r}), by Theorem 2.1 we have

Subcase 2.2. There exist vÎU, such that_{r}, the resulting graph is denoted by U′(r, 2, k) (see

pendent pendant edges in (U′(r,2,k)) belong to M, we know that

edges in M, they not all belong to E(C_{r}), by Lemma 2.2 and Theorem 2.2 we have

If

tion. So

“⇐” Case 1. Let

Theorem 2.1 we have

subgraph U(r, 1) (see

the r/2 edges in M, not all belong to E(C_{r}), by Theorem 2.2 we have

Case 2. Let

_{r}), by Theorem 2.1 we have

Let l = 1 in U(r, l, k+1) (

Theorem 3.3 The nullity set of U_{0,1}(n, r) is {0, 1, 2, ..., n-r-1}.

Proof. By Corollary 2.3, we only need to show that for each

graph

Case 1.

we get C_{r}, by Lemma 2.6 and 2.5 we have

ter (n − 2)/2 steps, we get a P_{2}, by Lemmas 2.6 we have

Case 2.

(r+1)/2 steps, we get kK_{1}, by Lemmas 2.3 we have

Case 3.

ing Lemma 2.6, after l/2 steps, we get

Theorem 3.4 The nullity set of U_{0,2}(n, r) is

Proof. Similar to Theorem 2.3, if

), where

where

If we take

Corollary 3.9 [_{n} is

This work is supported by the Natural Science Foundation of Qinghai Province (Grant No. 2011-Z-911).

Shengbiao Hu, (2014) A Note on the Nullity of Unicyclic Graphs. Applied Mathematics,05,1623-1631. doi: 10.4236/am.2014.510156