Further Results on the Nullity of Signed Graphs

Abstract

The nullity of a graph is the multiplicity of the eigenvalue zero in its spectrum. A signed graph is a graph with a sign attached to each of its edges. In this paper, we apply the coefficient theorem on the characteristic polynomial of a signed graph and give two formulae on the nullity of signed graphs with cut-points. As applications of the above results, we investigate the nullity of the bicyclic signed graph Γ(∞(𝑝,𝑞,𝑙)), obtain the nullity set of unbalanced bicyclic signed graphs, and thus determine the nullity set of bicyclic signed graphs.

1. Introduction

Let 𝐺=(𝑉,𝐸) be a simple graph with vertex set 𝑉=𝑉(𝐺)={𝑣₁,𝑣₂,…,𝑣_𝑛} and edge set 𝐸=𝐸(𝐺). The adjacency matrix 𝐴=𝐴(𝐺)=(𝑎_𝑖𝑗)_𝑛×𝑛 of 𝐺 is defined as follows: 𝑎_𝑖𝑗=1 if there exists an edge joining 𝑣_𝑖 and 𝑣_𝑗, and 𝑎_𝑖𝑗=0 otherwise. The nullity of a simple graph 𝐺 is the multiplicity of the eigenvalue zero in the spectrum of 𝐴(𝐺), denoted by 𝜂(𝐺). The rank of 𝐺 is referred to the rank of 𝐴(𝐺) and denoted by 𝑟(𝐺). Clearly, 𝜂(𝐺)+𝑟(𝐺)=𝑛 if 𝐺 has 𝑛 vertices.

A signed graph is a graph with a sign attached to each of its edges. Formally, a signed graph Γ(𝐺)=(𝐺,𝜎) consists of a simple graph 𝐺=(𝑉,𝐸) which is regarded as its underlying graph, and a mapping 𝜎∶𝐸→{+,−}, the edge labeling. To avoid confusion, we also write 𝑉(Γ(𝐺)) instead of 𝑉, 𝐸(Γ(𝐺)) instead of 𝐸, and 𝐸(Γ(𝐺))=𝐸^𝜎.

The adjacency matrix of Γ(𝐺) is 𝐴(Γ(𝐺))=(𝑎^𝜎_𝑖𝑗) with 𝑎^𝜎_𝑖𝑗=𝜎(𝑣_𝑖𝑣_𝑗)𝑎_𝑖𝑗, where (𝑎_𝑖𝑗) is the adjacency matrix of the underlying graph 𝐺. In the case of 𝜎=+, which is an all-positive edge labeling, 𝐴(𝐺,+) is exactly the classical adjacency matrix of 𝐺. So a simple graph is always assumed as a signed graph with all edges positive.

The nullity of a signed graph Γ(𝐺) is defined as the multiplicity of the eigenvalue zero in the spectrum of 𝐴(Γ(𝐺)), and is denoted by 𝜂(Γ(𝐺)). The rank of Γ(𝐺) is referred to the rank of 𝐴(Γ(𝐺)), and denoted by 𝑟(Γ(𝐺)). Surely, 𝜂(Γ(𝐺))+𝑟(Γ(𝐺))=𝑛 if Γ(𝐺) has 𝑛 vertices.

Let Γ(𝐺)=(𝐺,𝜎) be a signed graph and let Γ(𝐶) be a signed cycle of Γ(𝐺). The sign of Γ(𝐶) is defined by sgn(Γ(𝐶))=∏_{𝑒∈Γ(𝐶)}𝜎(𝑒). The cycle Γ(𝐶) is said to be positive or negative if sgn(Γ(𝐶))=+ or sgn(Γ(𝐶))=−. A signed graph is said to be balanced if all its cycles are positive, or equivalently, all cycles have even number of negative edges; otherwise it is called unbalanced.

About the nullity of simple graphs and its applications, there are many known results (see [1–8] for details). In 1953, Hurary [9] introduced the concept of a signed graph in connection with the study of the theory of social balance in social psychology. For more results of signed graphs and their applications, see [9–21].

In this paper, we obtain the coefficients theorem of the characteristic polynomial of a signed graph and give two formulae on the nullity of signed graphs with cut-points. As applications of the above results, we investigate the nullity of the bicyclic signed graph Γ(∞(𝑝,𝑞,𝑙)), obtain the nullity set of unbalanced bicyclic signed graphs, and thus determine the nullity set of bicyclic signed graphs.

2. The Coefficients of 𝑃_Γ(𝐺)(𝜆)

In this section, we obtain the coefficients theorem of the characteristic polynomial of a signed graph Γ(𝐺), 𝑃_Γ(𝐺)(𝜆), and the nullity of a signed cycle Γ(𝐶_𝑛) by using the coefficients theorem.

Theorem 1 (see [11]). Let 𝑃_𝐺(𝜆)=𝜆𝐼_𝑛−𝐴=𝜆^𝑛+𝑎₁𝜆^𝑛−1+⋯+𝑎_𝑛 (1) be the characteristic polynomial of an arbitrary undirected weighted multigraph 𝐺.
Call an “elementary figure”: (a) the graph 𝐾₂ or (b) every graph 𝐶_𝑞(𝑞≥1) (loops being included with 𝑞=1).
Call a basic figure 𝑈 if all of its components are elementary figures; let 𝑝(𝑈) and 𝑐(𝑈) be the number of components and the number of cycles contained in 𝑈, respectively, and let 𝒰_𝑖 denote the set of all basic figures contained in 𝐺 having exactly 𝑖 vertices. Then for any 𝑖∈{1,2,…,𝑛}, 𝑎_𝑖=_{𝑈∈𝒰𝑖}(−1)^𝑝(𝑈)⋅2^𝑐(𝑈)(𝑈),(𝑈)⋅=_{𝑒∈𝐸(𝑈)}(𝑤(𝑒))^{𝜉(𝑒;𝑈)}, (2) where 𝐸(𝑈) is the set of edges of 𝑈, 𝑤(𝑒) is the weight of the edge 𝑒, and 𝜉(𝑒;𝑈)=1,𝑖𝑓𝑒𝑖𝑠𝑐𝑜𝑛𝑡𝑎𝑖𝑛𝑒𝑑𝑖𝑛𝑠𝑜𝑚𝑒𝑐𝑦𝑐𝑙𝑒𝑜𝑓𝑈;2,𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒. (3)

Corollary 2. Let Γ(𝐺) be a signed graph on 𝑛 vertices and let 𝑃_Γ(𝐺)(𝜆)=𝜆𝐼_𝑛−𝐴(Γ(𝐺))=𝜆^𝑛+𝑎₁𝜆^𝑛−1+⋯+𝑎_𝑛 (4) be the characteristic polynomial of 𝐴(Γ(𝐺)). Then for any 𝑖∈{1,2,…,𝑛}, 𝑎_𝑖=_{𝒰Γ(𝑈)∈Γ𝑖}(−1)^{𝑝(Γ(𝑈))+𝑠(Γ(𝑈))}⋅2^{𝑐(Γ(𝑈))}, (5) where 𝑠(Γ(𝑈)) is the number of negative edges in cycle of Γ(𝑈); other notations are similar to Theorem 1.

Proof. Since Γ(𝐺) is a signed graph, 𝑤(𝑒)=+1 or −1, then ∏(Γ(𝑈))=∏_{𝑒∈𝐸(Γ(𝑈))}(−1)^{𝑠(Γ(𝑈))}. Thus the result follows from Theorem 1.

Proposition 3. (1) (see [11]). Let Γ(𝐶_𝑛) be a balanced cycle. Then 𝜂(Γ(𝐶_𝑛))=2 if 𝑛≡0(mod4) and 𝜂(Γ(𝐶_𝑛))=0 otherwise.
(2) (see [13]). Let Γ(𝐶_𝑛) be an unbalanced signed cycle. Then 𝜂(Γ(𝐶_𝑛))=2 if 𝑛≡2(mod4) and 𝜂(Γ(𝐶_𝑛))=0 otherwise.

Let 𝑠=𝑠(Γ(𝐶_𝑛)) be the number of negative edges of Γ(𝐶_𝑛). It is clear that Γ(𝐶_𝑛) is balanced if and only if 𝑠≡0(mod2); Γ(𝐶_𝑛) is unbalanced if and only if 𝑠≡1(mod2). Then Proposition 3 is equivalent to Theorem 4. We will give a new proof by Corollary 2.

Theorem 4. Let Γ(𝐶_𝑛)=(𝐶_𝑛,𝜎) be a signed cycle on 𝑛 vertices, and 𝑠 is the number of negative edges of Γ(𝐶_𝑛). Then 𝜂Γ𝐶_𝑛=()();()();();()();()().2,𝑖𝑓𝑛≡0mod4,𝑠≡0mod22,𝑖𝑓𝑛≡2mod4,𝑠≡1mod20,𝑖𝑓𝑛≡1mod20,𝑖𝑓𝑛≡0mod4,𝑠≡1mod20,𝑖𝑓𝑛≡2mod4,𝑠≡0mod2 (6)

Proof. By Corollary 2, for any 𝑖∈{1,2,…,𝑛}, we have 𝑎_𝑖=_{𝒰Γ(𝑈)∈Γ𝑖}(−1)^{𝑝(Γ(𝑈))+𝑠(Γ(𝑈))}⋅2^{𝑐(Γ(𝑈))}. (7)
Case 1. 𝑛≡1(mod2).
Clearly, Γ(𝑈)=Γ(𝐶_𝑛). Thus 𝑎_𝑛=2⋅(−1)^𝑠+1≠0.
Case 2. 𝑛≡0(mod2).
Clearly, Γ(𝑈)=Γ(𝐶_𝑛) or Γ(𝑈)=(𝑛/2)Γ(𝐾₂), and there exist two basic figures (𝑛/2)Γ(𝐾₂) in Γ(𝐶_𝑛). Then 𝑎_𝑛=2⋅(−1)^𝑠+1+2⋅(−1)^𝑛/2=()();()();()();()().0,if𝑛≡0mod4,𝑠≡0mod24,if𝑛≡0mod4,𝑠≡1mod2−4,if𝑛≡2mod4,𝑠≡0mod20,if𝑛≡2mod4,𝑠≡1mod2 (8) If 𝑎_𝑛≠0, then 𝜂(Γ(𝐶_𝑛))=0.
If 𝑎_𝑛=0, then we consider 𝑎_𝑛−1 and 𝑎_𝑛−2. Since 𝑛 is even, it is clear that 𝑎_𝑛−1=0 and 𝑎_𝑛−2≠0 by Corollary 2. Thus 𝜂(Γ(𝐶_𝑛))=2.

Similarly, by Corollary 2, we have the following.

Proposition 5. Let Γ(𝑃_𝑛)=(𝑃_𝑛,𝜎) be a signed path on 𝑛 vertices. Then 𝜂Γ𝑃_𝑛=1,𝑖𝑓𝑛𝑖𝑠𝑜𝑑𝑑;0,𝑖𝑓𝑛𝑖𝑠𝑒𝑣𝑒𝑛. (9)

3. The Nullity of a Signed Graph with Cut-Points

In this section, we deduce two concise formulae on the nullity of signed graphs with cut-points by similar technique applied in [3].

We first introduce some concepts and notations.

Let 𝐺 be a simple graph with vertex set 𝑉=𝑉(𝐺). For a nonempty subset 𝑈 of 𝑉, the subgraph with vertex set 𝑈 and edge set consisting of those pairs of vertices that are edges in 𝐺 is called the induced subgraph of 𝐺, denoted by 𝐺[𝑈]. Denote by 𝐺−𝑈, where 𝑈⊆𝑉, the graph obtained from 𝐺 by removing the vertices of 𝑈 together with all edges incident to them. Sometimes we use the notation 𝐺−𝐺₁ instead of 𝐺−𝑉(𝐺₁) if 𝐺₁ is an induced subgraph of 𝐺. For an induced subgraph 𝐺₁ (of 𝐺) and 𝑣∈𝐺−𝐺₁, the induced subgraph 𝐺[𝑉(𝐺₁)∪{𝑣}] is simply written as 𝐺₁+𝑣. The vertex 𝑣∈𝑉 is called a cut-point of 𝐺 if the resultant graph 𝐺−𝑣 is disconnected.

Let 𝐴 be the adjacency matrix of a graph 𝐺=(𝑉(𝐺),𝐸(𝐺)) on 𝑛 vertices. For 𝑈⊆𝑉(𝐺), 𝑊⊆𝑉(𝐺), denoted by 𝐴[𝑈,𝑊] is the submatrix of 𝐴 with rows corresponding to the vertices of 𝑈 and columns corresponding to the vertices of 𝑊. To simplify, the submatrix 𝐴[𝑈,𝑈] is written as 𝐴[𝑈]. For convenience, we usually write 𝐴[𝐺₁,𝐺₂] instead of the standard 𝐴[𝑉(𝐺₁),𝑉(𝐺₂)] for the two induced subgraphs 𝐺₁ and 𝐺₂ of 𝐺. In particular, denoted by 𝐴[𝑣,𝐺] is the row vector of 𝐴 corresponding to the vertex 𝑣 and by 𝐴[𝑣,𝐺_𝑖] is the subvector of 𝐴[𝑣,𝐺] corresponding to the vertices of 𝐺_𝑖. We refer to Cvetković et al. [11] for more terminologies and notations not defined here.

The following lemma is obvious.

Lemma 6. Let 𝐺=𝐺₁∪𝐺₂∪⋯∪𝐺_𝑠 and Γ(𝐺) be a signed graph, where 𝐺₁,𝐺₂,…,𝐺_𝑠 are the connected components of 𝐺. Then Γ(𝐺₁),Γ(𝐺₂),…,Γ(𝐺_𝑠) are the connected components of Γ(𝐺), 𝑟(Γ(𝐺))=∑^𝑠_𝑖=1𝑟(Γ(𝐺_𝑖)), and 𝜂(Γ(𝐺))=∑^𝑠_𝑖=1𝜂(Γ(𝐺_𝑖)).

Theorem 7. Let Γ(𝐺) be a connected signed graph on 𝑛 vertices, let 𝑣 be a cut-point of Γ(𝐺), and let Γ(𝐺₁),Γ(𝐺₂),…,Γ(𝐺_𝑠) be all the components of Γ(𝐺−𝑣). If there exists a component, say Γ(𝐺₁), among Γ(𝐺₁),Γ(𝐺₂),…,Γ(𝐺_𝑠) such that 𝜂(Γ(𝐺₁))=𝜂(Γ(𝐺₁+𝑣))+1, then 𝜂(Γ(𝐺))=𝜂(Γ(𝐺−𝑣))−1=∑^𝑠_𝑖=1𝜂(Γ(𝐺_𝑖))−1.

Proof. Let 𝐴=𝐴(Γ(𝐺)) be the adjacency matrix of Γ(𝐺). For each 𝑖, denote by 𝐴[Γ(𝐺_𝑖)] the adjacency matrix of the subgraph Γ(𝐺_𝑖) and by 𝐴[𝑣,Γ(𝐺_𝑖)] the subvector of 𝐴[𝑣,Γ(𝐺)] corresponding to the vertices of Γ(𝐺_𝑖); then the matrix 𝐴 can be partitioned as follows: 0𝐴𝑣,Γ𝐺₁𝐴𝑣,Γ𝐺₂⋯𝐴𝑣,Γ𝐺_𝑠𝐴Γ𝐺₁,𝑣𝐴Γ𝐺₁𝐴Γ𝐺0⋯0₂,𝑣0𝐴Γ𝐺₂𝐴Γ𝐺⋯0⋯⋯⋯⋯⋯_𝑠,𝑣00⋯𝐴Γ𝐺_𝑠, (10) where 𝐴[Γ(𝐺_𝑖),𝑣]=𝐴[𝑣,Γ(𝐺_𝑖)]^𝑇(𝑖=1,2,…,𝑠), the transpose of 𝐴[𝑣,Γ(𝐺_𝑖)], for each 𝑖.
Note that 𝜂(Γ(𝐺₁))=𝜂(Γ(𝐺₁+𝑣))+1, then 𝑟(Γ(𝐺₁+𝑣))=𝑟(Γ(𝐺₁))+2, thus the row vector 𝐴[𝑣,Γ(𝐺₁)] is not linear combination of the row vectors of 𝐴[Γ(𝐺₁)], therefore the row vector 𝐴[𝑣,Γ(𝐺)] is not linear combination of any other row vectors of 𝐴[Γ(𝐺)]. Since 𝐴 is a symmetric matrix, the column vector 𝐴[Γ(𝐺),𝑣] is not linear combination of any other column vectors of 𝐴, which implies that 𝑟(Γ(𝐺))=𝑟(Γ(𝐺−𝑣))+2. Then by Lemma 6, 𝜂(Γ(𝐺(Γ(𝐺(Γ((Γ())=𝑛−𝑟))=𝑛−𝑟𝐺−𝑣))−2=𝜂𝐺−𝑣))−1=^𝑠_𝑖=1𝜂Γ𝐺_𝑖−1. (11)

Theorem 8. Let Γ(𝐺) be a connected signed graph on 𝑛 vertices, let 𝑣 be a cut-point of Γ(𝐺), and let Γ(𝐺₁) be a component of Γ(𝐺−𝑣). If 𝜂(Γ(𝐺₁))=𝜂(Γ(𝐺₁+𝑣))−1, then 𝜂(Γ(𝐺))=𝜂(Γ(𝐺₁))+𝜂(Γ(𝐺−𝐺₁)).

Proof. Let 𝐴=𝐴(Γ(𝐺)) be the adjacency matrix of Γ(𝐺). Then 𝐴Γ𝐺𝐴=¹𝐴Γ𝐺₁0𝐴𝐺,𝑣𝑣,Γ₁0𝐴𝑣,Γ𝐺−𝐺₁Γ−𝑣0𝐴𝐺−𝐺₁−𝑣,𝑣𝐴Γ𝐺−𝐺₁−𝑣. (12)
Because 𝜂(Γ(𝐺₁))=𝜂(Γ(𝐺₁+𝑣))−1, 𝑟(Γ(𝐺₁+𝑣))=𝑟(Γ(𝐺₁)) and thus the row vector 𝐴[𝑣,Γ(𝐺₁+𝑣)]=[𝐴[𝑣,Γ(𝐺₁)]  0] is linear combination of the row vectors of 𝐴[Γ(𝐺₁),Γ(𝐺₁+𝑣)]. Similarly, the column vector 𝐴[Γ(𝐺₁+𝑣),𝑣] is linear combination of the column vectors of 𝐴[Γ(𝐺₁+𝑣),Γ(𝐺₁)]. Thus 𝐵 can be obtained from 𝐴 by row and column linear transformations, where 𝐴Γ𝐺𝐵=¹0000𝐴𝑣,Γ𝐺−𝐺₁Γ−𝑣0𝐴𝐺−𝐺₁−𝑣,𝑣𝐴Γ𝐺−𝐺₁−𝑣. (13) It is easy to see that 𝐵 is the adjacency matrix of the union of Γ(𝐺₁) and Γ(𝐺−𝐺₁). Then we have 𝑟(𝐴)=𝑟(𝐵), which implies that 𝜂(Γ(𝐺))=𝑛−𝑟(𝐴)=𝑛−𝑟(𝐵)=𝜂(𝐵)=𝜂(Γ(𝐺₁))+𝜂(Γ(𝐺−𝐺₁)).

4. The Nullity of the Bicyclic Signed Graph Γ(∞(𝑝,𝑞,𝑙))

Firstly, we introduce some definitions and notations which will be used in the following.

A bicyclic graph is a simple connected graph in which the number of edges equals the number of vertices plus one. There are two basic bicyclic graphs: ∞-graph and Θ-graph. An ∞-graph, denoted by ∞(𝑝,𝑞,𝑙), is obtained from two vertex-disjoint cycles 𝐶_𝑝 and 𝐶_𝑞 by connecting one vertex of 𝐶_𝑝 and one of 𝐶_𝑞 with a path 𝑃_𝑙 of length 𝑙−1 (in the case of 𝑙=1, identifying the above two vertices); and a Θ-graph, denoted by Θ(𝑝,𝑞,𝑙), is a union of three internally disjoint paths 𝑃_𝑝+1,𝑃_𝑞+1,𝑃_𝑙+1 of length 𝑝,𝑞,𝑙, respectively, with common end vertices, where 𝑝,𝑞,𝑙≥1 and at most one of them is 1. Observe that any bicyclic graph 𝐺 is obtained from an ∞-graph or a Θ-graph (possibly) by attaching trees to some of its vertices.

Lemma 9 (see [13]). Let Γ(𝐺) be a signed graph containing a pendant vertex, and let Γ(𝐻) be the induced subgraph of Γ(𝐺) obtained by deleting this pendant vertex together with the vertex adjacent to it. Then 𝜂(Γ(𝐺))=𝜂(Γ(𝐻)).

Theorem 10. Let 𝑝,𝑞,𝑙 be integers with 𝑝,𝑞≥3,𝑙≥1, and 𝐺=∞(𝑝,𝑞,𝑙).(1)If 𝑝 and 𝑞 are odd, then 𝜂(Γ(𝐺=Γ𝐶))0,𝑖𝑓𝑙𝑖𝑠𝑒𝑣𝑒𝑛;0,𝑖𝑓𝑙𝑖𝑠𝑜𝑑𝑑𝑎𝑛𝑑𝑠_𝑝−𝑠Γ𝐶_𝑞+𝑞−𝑝2();Γ𝐶≡0mod21,𝑖𝑓𝑙=1𝑎𝑛𝑑𝑠_𝑝−𝑠Γ𝐶_𝑞+𝑞−𝑝2();()Γ𝐶≡1mod2≥1,𝑖𝑓𝑙≥3𝑖𝑠𝑜𝑑𝑑𝑎𝑛𝑑𝑠_𝑝−𝑠Γ𝐶_𝑞+𝑞−𝑝2≡1(mod2). (14)(2)If 𝑝 and 𝑞 have different parities, without loss of generality, let 𝑝 be even; then 𝜂(Γ(𝐺=Γ𝐶))0,𝑖𝑓𝜂_𝑝Γ𝐶=0;1,𝑖𝑓𝜂_𝑝=2. (15)(3)If 𝑝 and 𝑞 are even, then 𝜂(Γ(𝐺=Γ𝐶))3,𝑖𝑓𝑙𝑖𝑠𝑜𝑑𝑑,𝜂_𝑝=𝜂Γ𝐶_𝑞Γ𝐶=2;1,𝑖𝑓𝑙𝑖𝑠𝑜𝑑𝑑,𝜂_𝑝⋅𝜂Γ𝐶_𝑞Γ𝐶=0;2,𝑖𝑓𝑙𝑖𝑠𝑒𝑣𝑒𝑛,𝜂_𝑝=2𝑜𝑟𝜂Γ𝐶_𝑞Γ𝐶=2;0,𝑖𝑓𝑙𝑖𝑠𝑒𝑣𝑒𝑛,𝜂_𝑝=𝜂Γ𝐶_𝑞=0. (16)

Proof. The proof is as follows.
Case 1 (both 𝑝 and 𝑞 are odd). By Corollary 2, we know 𝑎_𝑖=_{𝒰Γ(𝑈)∈Γ𝑖}(−1)^{𝑝(Γ(𝑈))+𝑠(Γ(𝑈))}⋅2^{𝑐(Γ(𝑈))}(𝑖=1,2,…𝑛). (17)
Subcase 1.1 (𝑙 is even). Consider 𝑎_𝑛=(−1)^{𝑠(Γ(𝐶𝑝))+𝑠(Γ(𝐶𝑞))+((𝑙+2)/2)}×2²+(−1)^{(((𝑝−1)/2)+((𝑞−1)/2)+(𝑙/2))}×2⁰≠0. (18) So 𝜂(Γ(𝐺))=0.
Subcase 1.2 (𝑙 is odd). 𝑎_𝑛=(−1)^{𝑠(Γ(𝐶𝑝))+((𝑙+𝑞)/2)}×2¹+(−1)^{𝑠(Γ(𝐶𝑞))+((𝑙+𝑝)/2)}×2¹,𝑎_𝑛=0,if𝑠Γ𝐶_𝑝−𝑠Γ𝐶_𝑞+𝑞−𝑝2();𝑎≡1mod2_𝑛≠0,if𝑠Γ𝐶_𝑝−𝑠Γ𝐶_𝑞+𝑞−𝑝2≡0(mod2). (19)
Clearly, if 𝑎_𝑛≠0, 𝜂(Γ(𝐺))=0; 𝑎_𝑛=0, 𝜂(Γ(𝐺))≥1. It is obvious that 𝑎_𝑛−1≠0 when 𝑙=1; thus 𝜂(Γ(𝐺))=1 when 𝑙=1.
Case 2 (𝑝 is even). Let 𝑣 be the vertex of Γ(𝐺) joining Γ(𝐶_𝑝) and Γ(𝑃_𝑙), then 𝑣 is a cut-point of Γ(𝐺). Note that 𝜂(Γ(𝐶_𝑝−𝑣))=𝜂(Γ(𝑃_𝑝−1))=1 by Proposition 5 and 𝜂(Γ(𝐶_𝑝))=2 or 0 by Theorem 4.
Subcase 2.1 (𝜂(Γ(𝐶_𝑝))=0). It is clear that 𝜂(Γ(𝐶_𝑝−𝑣))=𝜂(Γ(𝐶_𝑝))+1. Then by Theorem 7 and Lemma 9, 𝜂(Γ(𝐺(Γ(Γ𝐶))=𝜂𝐺−𝑣))−1=𝜂_𝑝−𝑣+𝜂Γ𝐺−𝐶_𝑝Γ−1=𝜂𝐺−𝐶_𝑝=𝜂Γ𝑃_𝑞−1𝜂Γ𝐶,if𝑙isodd;_𝑞,if𝑙iseven. (20)
Subcase 2.2 (𝜂(Γ(𝐶_𝑝))=2). It is clear that 𝜂(Γ(𝐶_𝑝−𝑣))=𝜂(Γ(𝐶_𝑝))−1. Then by Theorem 8 and Lemma 9, 𝜂(Γ(𝐺))=𝜂Γ𝐶_𝑝−𝑣+𝜂Γ𝐺−𝐶_𝑝Γ+𝑣=1+𝜂𝐺−𝐶_𝑝=Γ𝐶+𝑣1+𝜂_𝑞Γ𝑃,if𝑙isodd;1+𝜂_𝑞−1,if𝑙iseven. (21)
By Theorem 4, 𝜂Γ𝐶_𝑞=0,if𝑞isodd;0or2,if𝑞iseven. (22) Then by Proposition 5 and previous arguments, (2) and (3) hold.

5. The Nullity Set of Bicyclic Signed Graphs

Denoted by ℬ_𝑛 is the set of all bicyclic graphs on 𝑛 vertices. Obviously, ℬ_𝑛 consists of three types of graphs: first type denoted by 𝐵⁺_𝑛 is the set of those graphs each of which is an ∞-graph, ∞(𝑝,𝑞,𝑙), with trees attached when 𝑙>1; second type denoted by 𝐵_𝑛⁺⁺ is the set of those graphs each of which is an ∞-graph, ∞(𝑝,𝑞,𝑙), with trees attached when 𝑙=1; third type denoted by Θ_𝑛 is the set of those graphs each of which is a Θ-graph, Θ(𝑝,𝑞,𝑙), with trees attached. Then ℬ_𝑛=𝐵⁺_𝑛∪𝐵_𝑛⁺⁺∪Θ_𝑛.

Let Γ(ℬ_𝑛) be the set of all bicyclic signed graphs on 𝑛 vertices. Clearly, Γ(ℬ_𝑛)=Γ(𝐵⁺_𝑛)∪Γ(𝐵_𝑛⁺⁺)∪Γ(Θ_𝑛).

Let Γ(𝐺)=(𝐺,𝜎) be a signed graph on 𝑛 vertices. Suppose 𝜃∶𝑉(𝐺)→{+,−} is a sign function. Switching Γ(𝐺) by 𝜃 means forming a new signed graph Γ(𝐺)^𝜃=(𝐺,𝜎^𝜃) whose underlying graph is the same as 𝐺, but whose sign function is defined on an edge 𝑢𝑣 by 𝜎^𝜃(𝑢𝑣)=𝜃(𝑢)𝜎(𝑢𝑣)𝜃(𝑣). Note that switching does not change the signs or balance of the cycles of Γ(𝐺). If we define a diagonal signature matrix 𝐷^𝜃=diag(𝑑₁,𝑑₂,…,𝑑_𝑛) with 𝑑_𝑖=𝜃(𝑣_𝑖) for each 𝑣_𝑖∈𝑉(𝐺), then 𝐴(Γ(𝐺)^𝜃)=𝐷^𝜃𝐴(Γ(𝐺))𝐷^𝜃. Two graphs, Γ₁(𝐺) and Γ₂(𝐺) are called switching equivalent, denoted by Γ₁(𝐺)∼Γ₂(𝐺), if there exists a switching function 𝜃 such that Γ₂(𝐺)=Γ^𝜃₁(𝐺), or equivalently, 𝐴(Γ₂(𝐺))=𝐷^𝜃𝐴(Γ₁(𝐺))𝐷^𝜃.

Theorem 11 (see [16]). Let Γ(𝐺)=(𝐺,𝜎) be a signed graph. Then Γ(𝐺) is balanced if and only if Γ(𝐺)=(𝐺,𝜎)∼(𝐺,+).

Note that switching equivalence is a relation of equivalence, and two switching equivalent graphs have the same nullity. Therefore, when we discuss the nullity of signed graphs, we can choose an arbitrary representative of each switching equivalent class. If a signed graph is balanced, by Theorem 11, it is switching equivalent to one with all edges positive, that is, the underlying graph. Thus we only need to consider the case of unbalanced.

Hu et al. [6], Li et al. [7] characterize the maximal nullity of bicyclic graphs and determine the the nullity set of ℬ_𝑛. Recently, Fan et al. [14] characterize the maximal, the second maximal nullity of bicyclic signed graphs.

Theorem 12 (see [7]). Let 𝑛 be a positive integer, [0,𝑛]={0,1,2,…,𝑛}. Then(1)let 𝑛≥7, the nullity set of 𝐵⁺_𝑛 is [0,𝑛−6];(2)let 𝑛≥8, the nullity set of 𝐵_𝑛⁺⁺ is [0,𝑛−6];(3)let 𝑛≥6, the nullity set of Θ_𝑛 is [0,𝑛−4].

In Section 5.1–Section 5.3, we firstly obtain an upper bound of the nullity of bicyclic signed graphs in Γ(𝐵⁺_𝑛) and Γ(𝐵_𝑛⁺⁺), and then we obtain the nullity set of unbalanced bicyclic signed graphs in Γ(𝐵⁺_𝑛), Γ(𝐵_𝑛⁺⁺), Γ(Θ_𝑛), respectively, and determine the nullity set of (unbalanced) bicyclic signed graphs.

5.1. The Nullity Set of Unbalanced Bicyclic Signed Graphs in Γ(𝐵⁺_𝑛)

Theorem 13. Let 𝑛≥7, Γ(𝐺)∈Γ(𝐵⁺_𝑛). Then 𝜂(Γ(𝐺))≤𝑛−6.

Proof. Let 𝐺∈𝐵⁺_𝑛 be a bicyclic graph with trees attached on an ∞-graph, ∞(𝑝,𝑞,𝑙), where 𝑝,𝑞≥3,𝑙≥2.
Case 1 (𝑝,𝑞∈{3,4}). Consider the following subcases.
Subcase 1.1 (𝑝=𝑞=4). Note that 𝑝+𝑞+𝑙−2=6+𝑙≥9,if𝑙isodd;8,if𝑙iseven. (23) Then by (3) of Theorem 10, 𝑟(Γ(∞(4,4,𝑙)))≥6.
Clearly, ∞(4,4,𝑙) is an induced subgraph of 𝐺; then 𝑟(Γ(𝐺))≥𝑟(Γ(∞(4,4,𝑙)))≥6. Therefore 𝜂(Γ(𝐺))≤𝑛−6.
Subcase 1.2 (𝑝≠4 or 𝑞≠4). In this case, there must exist a graph 𝐻 on 6 vertices as an induced subgraph of 𝐺, where 𝐻=𝐻₁ or 𝐻=𝐻₂, shown in Figure 1. By Lemma 9 repeatedly we obtain 𝜂(Γ(𝐻₁))=𝜂(Γ(𝐻₂))=0, then 𝑟(Γ(𝐻₁))=𝑟(Γ(𝐻₂))=6. Thus 𝑟(Γ(𝐺))≥𝑟(Γ(𝐻))≥6 and 𝜂(Γ(𝐺))≤𝑛−6.
Case 2 (𝑝≥5 or 𝑞≥5). Without loss of generality, we assume that 𝑝≥5. There must exist a graph 𝐻₃ on 𝑝+1 vertices shown in Figure 1 as an induced subgraph of 𝐺. By Lemma 9 and Proposition 5, it is easy to check that 𝜂Γ𝐻₃=0,if𝑝isodd;1,if𝑝iseven. (24) Hence 𝑟Γ𝐻₃=𝑝+1,if𝑝isodd;𝑝,if𝑝iseven. (25) Since 𝑝≥5, 𝑟(Γ(𝐻₃))≥6. Then 𝑟(Γ(𝐺))≥𝑟(Γ(𝐻₃))≥6. Thus 𝜂(Γ(𝐺))≤𝑛−6.

Figure 1 

Theorem 14. Let 𝑛≥7. Then the nullity set of unbalanced bicyclic signed graphs in Γ(𝐵⁺_𝑛) is [0,𝑛−6].

Proof. It suffices to show that for each 𝑘∈[0,𝑛−6], there exists an unbalanced bicyclic signed graph Γ(𝐺)∈Γ(𝐵⁺_𝑛) such that 𝜂(Γ(𝐺))=𝑘.
Case 1   (𝑘=0). It is clear that there exists an unbalanced bicyclic signed graph Γ(𝐺)=Γ(∞(𝑝,𝑞,𝑙))∈Γ(𝐵⁺_𝑛) satisfying 𝜂(Γ(𝐺))=0 by Theorem 10, where 𝑝,𝑞≥3,𝑙≥2.
Case 2  (𝑘=𝑛−6). Let 𝐺=𝐺₁ shown in Figure 2, where Γ(𝐺₁) contains a balanced quadrangle and an unbalanced triangle. Thus 𝜂(Γ(𝐶₄))=2 by Theorem 4.
If 𝑛=7, then 𝐺=∞(3,4,2) and 𝜂(Γ(𝐺))=1=𝑛−6 by (2) of Theorem 10.
If 𝑛≥8, then by Lemmas 9 and 6, we have 𝜂(Γ(𝐺))=𝜂(Γ(𝑃₂)∪Γ(𝐶₄)∪(𝑛−8)Γ(𝐾₁))=𝜂(Γ(𝑃₂))+𝜂(Γ(𝐶₄))+(𝑛−8)𝜂(Γ(𝐾₁))=0+2+(𝑛−8)=𝑛−6.
Case 3   (1≤𝑘≤𝑛−7). Let 𝐺=𝐺₂ shown in Figure 2, where Γ(𝐺₂) contains two unbalanced triangles. By using Lemma 9 repeatedly, after [(𝑛−𝑘−4)/2] steps, we have 𝜂(Γ(𝐺=𝜂Γ𝑃))₂∪Γ𝐶₃∪𝑘Γ𝐾₁𝜂𝑃,if𝑛−𝑘isodd;2Γ₂∪𝑘Γ𝐾₁,if𝑛−𝑘iseven. (26)
Hence by Lemma 6, 𝜂(Γ(𝐺=𝜂Γ𝑃))₂+𝜂Γ𝐶₃+𝑘𝜂Γ𝐾₁Γ𝑃=0+0+𝑘=𝑘,if𝑛−𝑘isodd;2𝜂₂+𝑘𝜂Γ𝐾₁=0+𝑘=𝑘,if𝑛−𝑘iseven. (27)

Figure 2 

5.2. The Nullity Set of Unbalanced Bicyclic Signed Graphs in Γ(𝐵_𝑛⁺⁺)

Theorem 15. Let 𝑛≥8, Γ(𝐺)∈Γ(𝐵_𝑛⁺⁺). Then 𝜂(Γ(𝐺))≤𝑛−6.

Proof. Let 𝐺∈𝐵_𝑛⁺⁺ be a bicyclic graph with trees attached on an ∞-graph, ∞(𝑝,𝑞,1), where 𝑝,𝑞≥3.
Case 1 (𝑝,𝑞∈{3,4}). In this case, there must exist a graph 𝐻 on ℎ vertices as an induced subgraph of 𝐺, where 𝐻=𝐻₄,𝐻₅ with ℎ=6, or 𝐻=𝐻₆,𝐻₇,𝐻₈,𝐻₉ with ℎ=7, or 𝐻=𝐻₁₀,𝐻₁₁,𝐻₁₂ with ℎ=8 shown in Figure 3. By Lemma 9 repeatedly and Theorem 4, we obtain 𝜂(Γ(𝐻₄))=𝜂(Γ(𝐻₅))=𝜂(Γ(𝐻₈))=0, 𝜂(Γ(𝐻₆))=𝜂(Γ(𝐻₇))=𝜂(Γ(𝐻₉))=1, 𝜂(Γ(𝐻₁₁))=𝜂(Γ(𝐻₁₂))=2, and let Γ(𝐶₄) be the quadrangle containing no pendent edge of Γ(𝐻₁₀); we have 𝜂Γ𝐻₁₀=2,ifΓ𝐶₄𝐶isbalanced;0,ifΓ₄isunbalanced. (28)
Hence for each Γ(𝐻_𝑖)(𝑖=4,5,…,12), we have 𝑟(Γ(𝐻_𝑖))≥6, so 𝑟(Γ(𝐺))≥𝑟(Γ(𝐻_𝑖))≥6. Thus 𝜂(Γ(𝐺))≤𝑛−6.
Case 2 (𝑝≥5 or 𝑞≥5). Without loss of generality, we assume that 𝑝≥5. There must exist a graph 𝐻₃ on 𝑝+1 vertices shown in Figure 1 as an induced subgraph of 𝐺. Similar to the proof of Case 2 in Theorem 13, we have 𝑟(Γ(𝐻₃))≥6. Then 𝑟(Γ(𝐺))≥𝑟(Γ(𝐻₃))≥6 and thus 𝜂(Γ(𝐺))≤𝑛−6.

Figure 3 

Lemma 16 (see [14]). Let 𝐻₁₃ be a graph on 5 vertices as shown in Figure 4, and the two triangles of Γ(𝐻₁₃) have the same balance. Then 𝜂(Γ(𝐻₁₃))=𝑛−5=0.

Figure 4 

Theorem 17. Let 𝑛≥8; then the nullity set of unbalanced bicyclic signed graphs in Γ(𝐵_𝑛⁺⁺) is [0,𝑛−6].

Proof. It suffices to show that for each 𝑘∈[0,𝑛−6], there exists an unbalanced bicyclic signed graph Γ(𝐺)∈Γ(𝐵_𝑛⁺⁺) such that 𝜂(Γ(𝐺))=𝑘.
Case 1 (𝑘=0). It is clear that there exists an unbalanced bicyclic signed graph Γ(𝐺)=Γ(∞(𝑝,𝑞,1))∈Γ(𝐵_𝑛⁺⁺) satisfying 𝜂(Γ(𝐺))=0 by Theorem 10, where 𝑝,𝑞≥3.
Case 2 (𝑘=𝑛−6). Let 𝐺=𝐺₃ shown in Figure 4, where the two triangles of Γ(𝐺₃) are unbalanced. Then by Lemmas 9 and 6, we have 𝜂(Γ(𝐺))=𝜂(Γ(𝑃₂)∪(𝑛−6)Γ(𝐾₁))=𝜂(Γ(𝑃₂))+(𝑛−6)𝜂(Γ(𝐾₁))=0+(𝑛−6)=𝑛−6.
Case 3 (1≤𝑘≤𝑛−7). Let 𝐺=𝐺₄ shown in Figure 5, where the two triangles of Γ(𝐺₄) are unbalanced.
Subcase 3.1 (𝑛−𝑘 is odd). By using Lemma 9 repeatedly, after (𝑛−𝑘−5)/2 steps, we obtain the graph 𝑘Γ(𝐾₁)∪Γ(𝐻₁₃), where the two triangles of Γ(𝐻₁₃) are unbalanced. Hence 𝜂(Γ(𝐺))=𝜂(𝑘Γ(𝐾₁)∪Γ(𝐻₁₃))=𝜂(𝑘Γ(𝐾₁))+𝜂(Γ(𝐻₁₃))=𝑘+0=𝑘 by Lemma 16.
Subcase 3.2 (𝑛−𝑘 is even). By using Lemma 9 repeatedly, after (𝑛−𝑘−2)/2 steps, we obtain the graph 𝑘Γ(𝐾₁)∪Γ(𝑃₂). Hence 𝜂(Γ(𝐺))=𝜂(𝑘Γ(𝐾₁)∪Γ(𝑃₂))=𝜂(𝑘Γ(𝐾₁))+𝜂(Γ(𝑃₂))=𝑘+0=𝑘.

Figure 5 

5.3. The Nullity Set of Unbalanced Bicyclic Signed Graphs in Γ(Θ_𝑛)

Lemma 18 (see [14]). Let Γ(𝐺) be an unbalanced bicyclic signed graph on 𝑛 vertices. Then 𝜂(Γ(𝐺))≤𝑛−3, with equality if and only if Γ(𝐺)=Γ(Θ(2,2,1)) and the two triangles of Γ(Θ(2,2,1)) are both unbalanced.

By Lemma 18, we obtain the following result immediately.

Proposition 19. Let 𝑛≥5 and let Γ(𝐺) be an unbalanced bicyclic signed graph in Γ(Θ_𝑛). Then 𝜂(Γ(𝐺))≤𝑛−4.

Theorem 20. Let 𝑛≥6. Then the nullity set of unbalanced bicyclic signed graphs in Γ(Θ_𝑛) is [0,𝑛−4].

Proof. It suffices to show that for each 𝑘∈[0,𝑛−4], there exists an unbalanced bicyclic signed graph Γ(𝐺)∈Γ(Θ_𝑛) such that 𝜂(Γ(𝐺))=𝑘.
Case 1 (𝑘=0). Let 𝐺=𝐺₅ shown in Figure 6, where Γ(𝐺₅) contains at least an unbalanced triangle. By Lemma 9 and Theorem 4 (when 𝑛 is odd) or Proposition 5 (when 𝑛 is even), we have 𝜂(Γ(𝐺))=0.
Case 2 (𝑘=𝑛−4). Let 𝐺=𝐺₆ shown in Figure 6, where Γ(𝐺₆) contains at least an unbalanced triangle. By Lemma 9 and Proposition 5, we have 𝜂(Γ(𝐺))=𝜂(Γ(𝑃₃)∪(𝑛−5)Γ(𝐾₁))=𝜂(Γ(𝑃₃))+(𝑛−5)𝜂(Γ(𝐾₁))=1+(𝑛−5)=𝑛−4.
Case 3 (1≤𝑘≤𝑛−5). Consider the following subcases.
Subcase 3.1 (𝑛−𝑘 is odd). Let 𝐺=𝐺₇ shown in Figure 6, where the two triangles of 𝐺₇ are unbalanced. By using Lemma 9 repeatedly, after (𝑛−𝑘−3)/2 steps, we obtain the graph Γ(Θ(2,2,1))∪(𝑘−1)Γ(𝐾₁). Hence 𝜂(Γ(𝐺))=𝜂(Γ(Θ(2,2,1))∪(𝑘−1)Γ(𝐾₁))=𝜂(Γ(Θ(2,2,1)))+(𝑘−1)𝜂(Γ(𝐾₁))=1+(𝑘−1)=𝑘 by Lemmas 9 and 18.
Subcase 3.2 (𝑛−𝑘 is even). Let 𝐺=𝐺₈ shown in Figure 6, where the triangle of 𝐺₈ is unbalanced and the quadrangle is balanced. By using Lemma 9 repeatedly, after (𝑛−𝑘−2)/2 steps, we obtain the graph Γ(𝐶₄)∪(𝑘−2)Γ(𝐾₁). Hence 𝜂(Γ(𝐺))=𝜂(Γ(𝐶₄)∪(𝑘−2)Γ(𝐾₁))=𝜂(Γ(𝐶₄))+(𝑘−2)𝜂(Γ(𝐾₁))=2+(𝑘−2)=𝑘 by Lemma 9 and Theorem 4.

Figure 6 

5.4. In Conclusion

From the above discussion, by Theorems 12, 14, 17 and 20 we can obtain the following results immediately.

Theorem 21. Let 𝑛≥8. Then the nullity set of unbalanced bicyclic signed graphs is [0,𝑛−4].

Theorem 22. Let 𝑛≥8. Then the nullity set of bicyclic signed graphs is [0,𝑛−4].

When 4≤𝑛≤7, the nullity set of bicyclic signed graphs is easy to obtain by known results and direct calculation, so we omit it.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors would like to thank the referees for their valuable comments, corrections, and suggestions, which lead to an improvement of the original paper. Research was supported by National Natural Science Foundation of China (no. 10901061), the Zhujiang Technology New Star Foundation of Guangzhou (no. 2011J2200090), and Program on International Cooperation and Innovation, Department of Education, Guangdong Province (no. 2012gjhz0007).