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No graph with nullity η(G)=|V(G)|−2m(G)+2c(G)−1
Discrete Applied Mathematics 268 (2019) 130–136
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Discrete Applied Mathematics journal homepage: www.elsevier.com/locate/dam
No graph with nullity η(G) = |V (G)| − 2m(G) + 2c(G) − 1✩ Xin Li, Ji-Ming Guo
∗
Department of Mathematics, East China University of Science and Technology, Shanghai, PR China
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Article history: Received 1 November 2018 Received in revised form 17 April 2019 Accepted 17 April 2019 Available online 16 May 2019 Keywords: Adjacency matrix Eigenvalue Nullity Graph
a b s t r a c t Let G = (V (G), E(G)) be a simple graph with vertex set V (G) and edge set E(G), c(G) = |E(G)| − |V (G)| + θ (G) be the dimension of cycle spaces of G with θ (G) the number of connected components of G, m(G) be the matching number of G, and η(G) be the nullity of G. It was shown in Wang and Wong (2014) that the nullity η(G) of G is bounded by an upper bound and a lower bound as
|V (G)| − 2m(G) − c(G) ≤ η(G) ≤ |V (G)| − 2m(G) + 2c(G). Graphs with nullity attaining the upper bound and the lower bound have been characterized by Song et al. (2015) and Wang (2016), respectively. In this paper, we prove that there is no graph with nullity η(G) = |V (G)| − 2m(G) + 2c(G) − 1. For fixed c(G), infinitely many connected graphs with nullity |V (G)| − 2m(G) + 2c(G) − k, (0 ≤ k ≤ 3c(G), k ̸ = 1) are also constructed. © 2019 Elsevier B.V. All rights reserved.
1. Introduction Throughout this paper we consider simple undirected graphs, i.e., without loops and multiple edges. Let G = (V , E) be a simple graph with vertex set V = {v1 , v2 , . . . , vn } and edge set E. Its adjacency matrix A(G) = (aij ) is defined as n × n matrix (aij ), where aij = 1, if vi is adjacent to vj ; and aij = 0, otherwise. Let |V (G)| (|E(G)|) be the number of vertices (edges) of G, and d(v ) be the degree of the vertex v ∈ V (G). The nullity η(G) is defined to be the multiplicity of 0 as an eigenvalue of A(G). The dimension of cycle spaces of G, written as c(G), is defined as c(G) = |E(G)| − |V (G)| + θ (G), where θ (G) denotes the number of connected components of G. When G is connected, then G is a tree if c(G) = 0; G is a unicyclic graph if c(G) = 1; G is a bicyclic graph if c(G) = 2; and G is a tricyclic graph if c(G) = 3. A graph H is a subgraph of G if V (H) ⊆ V (G) and E(H) ⊆ E(G). Further, H is called an induced subgraph of G if two vertices of V (H) are adjacent in H if and only if they are adjacent in G; H is called a spanning subgraph of G if and only if V (H) = V (G) and E(H) ⊆ E(G). For a subset U of V (G), denote by G − U the graph obtained from G by removing the vertices of U together with all edges incident to them. By Cn (resp., Pn , K1,n−1 and K1 ) we denote a cycle (resp., a path, a star with n vertices and an isolated vertex). A pendant vertex is a vertex with degree one. A cycle is attaching at a pendant vertex as an induced subgraph of G, that is the cycle has a unique vertex of degree 3 and the other vertices with degree 2, is called a pendant cycle of G. The length of a path Pk is the number of edges on Pk , and the distance between vertex u and v , denoted by d(u, v ), is the length of the shortest path between u and v . The girth of G is the length of a shortest cycle in G, denoted by g(G). A matching M in G is a set of pairwise non-adjacent edges. A maximum matching M(G) is a matching of G that contains ✩ This research is supported by NSFC, PR china (No. 11371372). ∗ Corresponding author. E-mail addresses: [email protected] (X. Li), [email protected] (J.-M. Guo). https://doi.org/10.1016/j.dam.2019.04.018 0166-218X/© 2019 Elsevier B.V. All rights reserved.
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Fig. 1. Graphs ∞(p, q, l) and θ (p, q, l).
Fig. 2. Basic tricyclic graphs.
the largest possible number of edges. The matching number of G, denoted by m(G), is the size of a maximum matching. A matching M saturates a vertex v , and v is said to be M-saturated, if some edge of M is incident with v ; otherwise, v is M-unsaturated. Let M be a matching in G. An M-alternating path in G is a path whose edges are alternately in E \M (the set of edges belong to E, but not M) and M. An M-augmenting path is an M-alternating path whose origin and terminus are M-unsaturated. There are two basic bicyclic graphs [16]: ∞-graph and θ -graph, which are depicted in Fig. 1. An ∞-graph, denoted by ∞(p, q, l), is obtained from two vertex-disjoint cycles Cp and Cq by connecting some vertex of Cp and some vertex of Cq with a path 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 Pp+1 , Pq+1 , Pl+1 of length p, q, l resp. with common end vertices, where p, q, l ≥ 1 and at most one of them is 1. Note that any bicyclic graph is obtained from an ∞-graph or a θ -graph by attaching trees to some of its vertices. Similarly, for any tricyclic graph G, the base of G, denoted by GB , is the minimal tricyclic subgraph of G. Clearly GB is the unique tricyclic of G containing no pendant vertex, and G can be obtained from GB by attaching trees to some vertices of GB . It follows from [8] that there are eight types of bases for tricyclic graph, which are depicted in Fig. 2. In [5], Collatz and Sinogowitz first posed the problem of characterizing all graphs that satisfy η(G) > 0. This question is of great interest in chemistry, because η(G) = 0 is a necessary (but not sufficient) condition for a so-called conjugated molecule to be chemically stable, where G is the graph representing the carbon-atom skeleton of this molecule; The nullity of a graph is also important in mathematics, since it is related to the singularity of A(G). In recent years, this problem has been paid much attention by many researchers (see [3,4,9,11,12] and the references therein). It was shown in [15] that the nullity η(G) of G is bounded by an upper bound and a lower bound as
|V (G)| − 2m(G) − c(G) ≤ η(G) ≤ |V (G)| − 2m(G) + 2c(G). Graphs with nullity attaining the upper bound and the lower bound have been characterized by Song et al. [13] and Wang [14], respectively. In this paper, we prove that there is no graph with nullity η(G) = |V (G)| − 2m(G) + 2c(G) − 1.
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Furthermore, for fixed c(G), infinitely many connected graphs with nullity |V (G)|− 2m(G) + 2c(G) − k, (0 ≤ k ≤ 3c(G), k ̸ = 1) are also constructed. 2. Lemmas and results By the well known Cauchy-interlacing theorem, we have the following. Lemma 1 ([2]). If x is a vertex of a graph G, then η(G) − 1 ≤ η(G − x) ≤ η(G) + 1. Lemma 2 ([15]). For any graph G, |V (G)| − 2m(G) − c(G) ≤ η(G) ≤ |V (G)| − 2m(G) + 2c(G). Let G be a graph with pairwise vertex-disjoint cycles (if any), and let C (G) denote the set of all cycles in G. The acyclic graph TG is obtained from G by contracting cycles. More specifically, the vertex set V (TG ) is taken to be U ∪ WG , where U consists of all vertices of G that do not lie on a cycle and WG = {vC : C ∈ C (G)}. Two vertices in U are adjacent in TG if and only if they are adjacent in G, a vertex u ∈ U is adjacent to a vertex vC ∈ WG if and only if u is adjacent (in G) to a vertex in the cycle C . It is clear that TG is always acyclic and is a tree when G is connected. The following result determined all the graphs with nullity attaining the upper bound of Lemma 2. Lemma 3 ([13]). For any graph G, η(G) = |V (G)| − 2m(G) + 2c(G) if and only if the following three conditions are all satisfied: (a) Distinct cycles of G (if any) are vertex-disjoint; (b) The length of each cycle of G (if any) is a multiple of 4; (c) m(TG ) = m(TG − WG ) or, equivalently, there exists a maximum matching of TG that does not cover any vertex in WG . For convenience, hereafter in this paper we say a graph G satisfies the maximal nullity condition if η(G) = |V (G)| − 2m(G) + 2c(G). Definition 1. Let G be a graph with at least one pendant vertex. The operation of deleting a pendant vertex and its adjacent vertex from G is called PED (short form for pendant edge deletion). Lemma 4 ([6]). Let G be a graph and H be the graph obtained from G by PED operation. Then η(G) = η(H). Lemma 5 ([7]). For every acyclic graph T with at least one vertex, η(T ) = |V (T )| − 2m(T ). Lemma 6 ([7]). η(Pn ) equals 1 if n is odd and equals 0 if n is even; η(Cn ) equals 2 if n is a multiple of 4 and equals 0, otherwise. Lemma 7 ([10]). Suppose G is a unicyclic graph with n vertices and the cycle in G is Cl . Let E1 be the set of edges of G between Cl and G − Cl and E2 the set of matchings of m(G) edges. Then (1) η(G) = n − 2m(G) − 1 if m(G) = (l − 1)/2 + m(G − Cl ); (2) η(G) = n − 2m(G) + 2 if G satisfies properties: m(G) = l/2 + m(G − Cl ), l = 0(mod 4) and E1 ∩ M = ∅ for all M ∈ E2 ; (3) η(G) = n − 2m(G) otherwise. Remark: From Lemma 7, there exist unicyclic graphs with nullity n − 2m(G) − 1 = |V (G)|− 2m(G) − c(G), n − 2m(G) and n − 2m(G) + 2, respectively. The unicyclic graph does not exist only for nullity n − 2m(G) + 1 = |V (G)| − 2m(G) + 2c(G) − 1. An induced subgraph S of a graph G is called a pendant star of G if S is a star with at least two vertices such that the center of S is the only vertex which is adjacent to some vertices not in S. Lemma 8 ([10]). If S is a pendant star of a graph G, then m(G) = 1 + m(G − S). Furthermore, G has a maximum matching which consists of edges in a maximum matching for G − S together with an edge of S. From Lemma 8, we immediately have the following corollary: Corollary 1.
Let G be a graph with a pendant vertex u, and uv ∈ E(G). Then m(G) = m(G − u − v ) + 1.
Lemma 9 ([1]). A matching M of graph G is a maximum matching if and only if G contains no M-augmenting path.
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Lemma 10. Let G be a graph with a pendant vertex u, uv ∈ E(G) and v does not lie on any cycle of G. Then η(G) = |V (G)| − 2m(G) + 2c(G) − k, (0 ≤ k ≤ 3c(G)) if and only if η(G − u − v ) = |V (G − u − v )| − 2m(G − u − v ) + 2c(G − u − v ) − k. Proof. It is easy to get
|V (G)| = |V (G − u − v )| + 2.
(1)
|E(G)| = |E(G − u − v )| + d(v ).
(2)
and
θ (G − u − v ) = θ (G) + d(v ) − 2.
(3)
By Eqs. (1)–(3), we can get c(G) = |E(G)| − |V (G)| + θ (G)
= |E(G − u − v )| + d(v ) − |V (G − u − v )| − 2 + θ (G − u − v ) − d(v ) + 2 = |E(G − u − v )| − |V (G − u − v )| + θ (G − u − v ) = c(G − u − v ).
(4)
Then by Eqs. (1), (4), Corollary 1 and Lemma 4, we have
η(G − u − v ) = η(G) = |V (G)| − 2m(G) + 2c(G) − k = |V (G − u − v )| + 2 − 2(m(G − u − v ) + 1) + 2c(G − u − v ) − k = |V (G − u − v )| − 2m(G − u − v ) + 2c(G − u − v ) − k. □ Lemma 11. Let G be a graph with a pendant vertex u, and uv ∈ E(G). If v lies on some cycle of G, then η(G) ≤ |V (G)| − 2m(G) + 2c(G) − 2. Proof. We prove the result by contradiction. From Lemma 2, we can suppose that
η(G) = |V (G)| − 2m(G) + 2c(G) − k, k = 0, 1.
(5)
Note that v lies on some cycle of G, then
θ (G − u − v ) ≤ θ (G) + d(v ) − 3. Thus we have c(G) = |E(G)| − |V (G)| + θ (G)
≥ |E(G − u − v )| + d(v ) − |V (G − u − v )| − 2 + θ (G − u − v ) − d(v ) + 3 = |E(G − u − v )| − |V (G − u − v )| + θ (G − u − v ) + 1 = c(G − u − v ) + 1.
(6)
Then by Eqs. (5), (6), Corollary 1 and Lemma 4, we have
η(G − u − v ) = η(G) = |V (G)| − 2m(G) + 2c(G) − k ≥ |V (G − u − v )| + 2 − 2(m(G − u − v ) + 1) + 2(c(G − u − v ) + 1) − k = |V (G − u − v )| − 2m(G − u − v ) + 2c(G − u − v ) + 2 − k (k = 0, 1), a contradiction to Lemma 2.
□
Let F be the graph in Fig. 3. For any vertex v of graph F , it is easy to see that F −v is either K1,3 or C4 . Let D = F ∪(n−5)K1 . Then by Lemmas 3 and 5, we have η(D) ̸ = |V (D)| − 2m(D) + 2c(D) but η(D − v ) = |V (D − v )| − 2m(D − v ) + 2c(D − v ). Lemma 12. Let G ̸ = D be a graph without pendant vertices. If η(G) ̸ = |V (G)| − 2m(G) + 2c(G) and c(G) ≥ 2, then there exists a vertex, say x, on some cycle of G such that η(G − x) ̸ = |V (G − x)| − 2m(G − x) + 2c(G − x). Proof. If g(G) = 3, and c(G) ≥ 2, then there must exist a vertex, say x, on some cycle of G such that G − x contains C3 as an induced subgraph. From Lemma 3, we have η(G − x) ̸ = |V (G − x)| − 2m(G − x) + 2c(G − x). In the following we suppose that g(G) ≥ 4. Since η(G) ̸ = |V (G)| − 2m(G) + 2c(G), G does not satisfy at least one of the three cases in Lemma 3. Without loss of generality, we distinguish the following three cases.
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Fig. 3. Graph F .
Case 1. Suppose that G does not satisfy case (a) of Lemma 3. Then there exist at least two vertex-joint cycles, say Ck (k ≥ 4) and Cs (s ≥ 4), in G. Let G(Ck , Cs ) be the subgraph induced by Ck and Cs . The following two subcases can be identified for this case. Subcase 1.1. c(G) = 2. Then G is the union of G(Ck , Cs ) and some isolated vertices (if any), and G(Ck , Cs ) is either ∞-graph (l = 1) or θ -graph (see Fig. 1). Since G ̸= D, there exists a vertex, say x, on some cycle of G such that G − x has a cycle with a pendant edge. By Lemma 11, we have η(G − x) ̸ = |V (G − x)| − 2m(G − x) + 2c(G − x). Subcase 1.2. c(G) ≥ 3. If there exists a vertex x on some cycle of G and x ∈ / G(Ck , Cs ), then η(G − x) ̸= |V (G − x)| − 2m(G − x) + 2c(G − x) by Lemma 3; otherwise, G(Ck , Cs ) contains one of tricyclic graphs e, f , g , h (see Fig. 2) as a subgraph. Then there exists a vertex, say x on some cycle of G such that G − x has two vertex-joint cycles. Then from Lemma 3, we have η(G − x) ̸ = |V (G − x)| − 2m(G − x) + 2c(G − x). Case 2. Suppose G does not satisfy case (b) but satisfies case (a) of Lemma 3. Then there exists at least one cycle with length is not a multiple of 4. For c(G) ≥ 2, there exists a vertex x on another cycle of G such that η(G − x) ̸ = |V (G − x)| − 2m(G − x) + 2c(G − x) by Lemma 3. Case 3. Suppose G does not satisfy case (c) but satisfies cases (a) and (b) of Lemma 3. If E(TG ) = ∅, then G is the union of some cycles and some isolated vertices. So m(TG ) = m(TG − WG ) = 0, a contradiction to the assumption. Suppose E(TG ) ̸ = ∅ in the following. For every maximum matching M of TG , M must contain at least one pendent edge of TG . Otherwise we can find an M-augmenting path in TG , a contradiction with Lemma 9. Let u be a pendent vertex of TG . Since G has no pendant vertex, u ∈ WG . It is easy to see that the cycle in G corresponding to u is a pendant cycle, denoted by Cu . Let v be the unique vertex of Cu adjacent to a vertex outside Cu and x (x ̸ = v ) be a vertex of Cu . Then TG−x is the graph obtained from TG and Cu − x by identifying u and v as one vertex. The following two subcases can be identified for this case. Subcase 3.1. Suppose that every maximum matching of TG must contain all pendant edges of TG . Let x be one of two vertices of Cu such that x is adjacent to v . Note that the length of Cu is a multiple of 4. Then m(TG−x ) = m(TG ) + m(Cu − x) and M(TG ) ∪ M(Cu − x − v ) is a maximum matching of TG−x , where M(TG ) and M(Cu − x − v ) are the maximum matchings of TG and Cu − x − v , respectively. It is easy to see that M(TG ) ∪ M(Cu − x − v ) covers some vertex in WG−x . Then η(G − x) ̸= |V (G − x)| − 2m(G − x) + 2c(G − x) by Lemma 3. Subcase 3.2. Otherwise. There exist some pendant edge, say w u, and some maximum matching, say M(TG ), of TG such that w u ∈ / M(TG ). Let y be a vertex of Cu such that d(v, y) = 2. Then m(TG−y ) = m(TG ) + m(Cu − y) and M(TG ) ∪ M(Cu − y) is a maximum matching of TG−y , where M(Cu − y) is the maximum matching of Cu − y. It is easy to see that M(TG ) ∪ M(Cu − y) covers some vertex in WG−y . Then η(G − y) ̸ = |V (G − y)| − 2m(G − y) + 2c(G − y) by Lemma 3. □ Theorem 2.1.
Let G be a graph without pendant vertices. Then we have
η(G) ̸= |V (G)| − 2m(G) + 2c(G) − 1. Proof. If G = D, it is easy to see that η(G) ̸ = |V (G)| − 2m(G) + 2c(G) − 1. Suppose that G ̸ = D in the following. We prove the result by induction on c(G). If c(G) = 0, then G = nK1 , the result is obvious. If c(G) = 1, then G = Ck ∪ (n − k)K1 , (3 ≤ k ≤ n). By Lemma 6, the result follows. Now suppose that the result holds for c(G) ≤ k, (k ≥ 1). We next need to prove that the result holds for c(G) = k + 1. Suppose there exists some graph G with c(G) = k + 1, and
η(G) = |V (G)| − 2m(G) + 2c(G) − 1.
(7)
For the graph G − x, x is any vertex on the cycle of G, we have m(G) ≤ m(G − x) + 1.
(8)
θ (G − x) ≤ θ (G) + d(x) − 2.
(9)
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Fig. 4. G(k1 , k2 , k3 ).
By Eq. (9), we can get c(G) = |E(G)| − |V (G)| + θ (G)
≥ |E(G − x)| + d(x) − (|V (G − x)| + 1) + θ (G − x) − d(x) + 2 = c(G − x) + 1.
(10)
Then by Eqs. (7), (8) and (10), we have
η(G) = |V (G)| − 2m(G) + 2c(G) − 1 ≥ |V (G − x)| + 1 − 2(m(G − x) + 1) + 2(c(G − x) + 1) − 1 = |V (G − x)| − 2m(G − x) + 2c(G − x).
(11)
By Eq. (10), c(G − x) ≤ k, where x is any vertex on the cycle of G. Then by the assumption for induction, we have η(G − x) ̸= |V (G − x)| − 2m(G − x) + 2c(G − x) − 1. From Lemma 12, there exists some vertex, say y on the cycle of G, such that η(G − y) ̸ = |V (G − y)| − 2m(G − y) + 2c(G − y). So we have η(G − y) ≤ |V (G − y)| − 2m(G − y) + 2c(G − y) − 2. Then by Lemma 1, we have η(G) ≤ η(G − y) + 1 ≤ |V (G − y)| − 2m(G − y) + 2c(G − y) − 1, a contradiction to Eq. (11). □ Now we give the main result of this paper. Theorem 2.2. For any graph G, η(G) ̸ = |V (G)| − 2m(G) + 2c(G) − 1. Proof. If G is an acyclic graph, by Lemma 5, the result follows. In the following, we suppose that G is not an acyclic graph. We prove the result by contradiction. Let G be not an acyclic graph with nullity η(G) = |V (G)| − 2m(G) + 2c(G) − 1. If G has a pendant edge, say uv ∈ E(G), where u is a pendant vertex and v does not lie on any cycle of G. By applying PED operation to G, we have η(G − u − v ) = |V (G − u − v )| − 2m(G − u − v ) + 2c(G − u − v ) − 1 from Lemma 10. Going on the above process at last we can get a graph H such that either H has no pendant vertices or H has a pendant edge, say wr, where r is a pendant vertex and w lies on some cycle of H, and η(H) = |V (H)| − 2m(H) + 2c(H) − 1, a contradiction to Theorem 2.1 or Lemma 11. □ Let G(k1 , k2 , k3 ) (ki ≥ 0, i = 1, 2, 3) be the connected graph obtained from k1 C3 , (k2 + k3 )C4 and K1,k3 +1 by adding k1 + k2 + k3 new edges among them (see Fig. 4). Theorem 2.3. For fixed c(G), there are infinitely many connected graphs with the nullity |V (G)| − 2m(G) + 2c(G) − k, (0 ≤ k ≤ 3c(G), k ̸ = 1). Proof. Let G(k1 , k2 , k3 ) (k1 + k2 + k3 = c(G(k1 , k2 , k3 ))) be the graph defined as Fig. 4. It is easy to see that
|V (G(k1 , k2 , k3 ))| = 3k1 + 4k2 + 5k3 + 2; m(G(k1 , k2 , k3 )) = k1 + 2k2 + 2k3 + 1; c(G(k1 , k2 , k3 )) = k1 + k2 + k3 .
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Since η(K1,2 ) = 1, η(C4 ) = 2, η(C3 ) = 0, from Lemma 4 and the above three equations, we have
η(G(k1 , k2 , k3 )) = η(G(k1 , k2 , k3 ) − u − v ) (by Lemma 4) = 2k2 + k3 = |V (G(k1 , k2 , k3 ))| − 2m(G(k1 , k2 , k3 )) + 2c(G(k1 , k2 , k3 )) − 3k1 − 2k3 . It is easy to see that 3k1 + 2k3 can take over every integer from zero to 3c(G) except for one. From Lemma 10, the result follows. □ References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16]
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No graph with nullity η(G) = |V (G)| − 2m(G) + 2c(G) − 1✩ Xin Li, Ji-Ming Guo
∗
Department of Mathematics, East China University of Science and Technology, Shanghai, PR China
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Article history: Received 1 November 2018 Received in revised form 17 April 2019 Accepted 17 April 2019 Available online 16 May 2019 Keywords: Adjacency matrix Eigenvalue Nullity Graph
a b s t r a c t Let G = (V (G), E(G)) be a simple graph with vertex set V (G) and edge set E(G), c(G) = |E(G)| − |V (G)| + θ (G) be the dimension of cycle spaces of G with θ (G) the number of connected components of G, m(G) be the matching number of G, and η(G) be the nullity of G. It was shown in Wang and Wong (2014) that the nullity η(G) of G is bounded by an upper bound and a lower bound as
|V (G)| − 2m(G) − c(G) ≤ η(G) ≤ |V (G)| − 2m(G) + 2c(G). Graphs with nullity attaining the upper bound and the lower bound have been characterized by Song et al. (2015) and Wang (2016), respectively. In this paper, we prove that there is no graph with nullity η(G) = |V (G)| − 2m(G) + 2c(G) − 1. For fixed c(G), infinitely many connected graphs with nullity |V (G)| − 2m(G) + 2c(G) − k, (0 ≤ k ≤ 3c(G), k ̸ = 1) are also constructed. © 2019 Elsevier B.V. All rights reserved.
1. Introduction Throughout this paper we consider simple undirected graphs, i.e., without loops and multiple edges. Let G = (V , E) be a simple graph with vertex set V = {v1 , v2 , . . . , vn } and edge set E. Its adjacency matrix A(G) = (aij ) is defined as n × n matrix (aij ), where aij = 1, if vi is adjacent to vj ; and aij = 0, otherwise. Let |V (G)| (|E(G)|) be the number of vertices (edges) of G, and d(v ) be the degree of the vertex v ∈ V (G). The nullity η(G) is defined to be the multiplicity of 0 as an eigenvalue of A(G). The dimension of cycle spaces of G, written as c(G), is defined as c(G) = |E(G)| − |V (G)| + θ (G), where θ (G) denotes the number of connected components of G. When G is connected, then G is a tree if c(G) = 0; G is a unicyclic graph if c(G) = 1; G is a bicyclic graph if c(G) = 2; and G is a tricyclic graph if c(G) = 3. A graph H is a subgraph of G if V (H) ⊆ V (G) and E(H) ⊆ E(G). Further, H is called an induced subgraph of G if two vertices of V (H) are adjacent in H if and only if they are adjacent in G; H is called a spanning subgraph of G if and only if V (H) = V (G) and E(H) ⊆ E(G). For a subset U of V (G), denote by G − U the graph obtained from G by removing the vertices of U together with all edges incident to them. By Cn (resp., Pn , K1,n−1 and K1 ) we denote a cycle (resp., a path, a star with n vertices and an isolated vertex). A pendant vertex is a vertex with degree one. A cycle is attaching at a pendant vertex as an induced subgraph of G, that is the cycle has a unique vertex of degree 3 and the other vertices with degree 2, is called a pendant cycle of G. The length of a path Pk is the number of edges on Pk , and the distance between vertex u and v , denoted by d(u, v ), is the length of the shortest path between u and v . The girth of G is the length of a shortest cycle in G, denoted by g(G). A matching M in G is a set of pairwise non-adjacent edges. A maximum matching M(G) is a matching of G that contains ✩ This research is supported by NSFC, PR china (No. 11371372). ∗ Corresponding author. E-mail addresses: [email protected] (X. Li), [email protected] (J.-M. Guo). https://doi.org/10.1016/j.dam.2019.04.018 0166-218X/© 2019 Elsevier B.V. All rights reserved.
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Fig. 1. Graphs ∞(p, q, l) and θ (p, q, l).
Fig. 2. Basic tricyclic graphs.
the largest possible number of edges. The matching number of G, denoted by m(G), is the size of a maximum matching. A matching M saturates a vertex v , and v is said to be M-saturated, if some edge of M is incident with v ; otherwise, v is M-unsaturated. Let M be a matching in G. An M-alternating path in G is a path whose edges are alternately in E \M (the set of edges belong to E, but not M) and M. An M-augmenting path is an M-alternating path whose origin and terminus are M-unsaturated. There are two basic bicyclic graphs [16]: ∞-graph and θ -graph, which are depicted in Fig. 1. An ∞-graph, denoted by ∞(p, q, l), is obtained from two vertex-disjoint cycles Cp and Cq by connecting some vertex of Cp and some vertex of Cq with a path 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 Pp+1 , Pq+1 , Pl+1 of length p, q, l resp. with common end vertices, where p, q, l ≥ 1 and at most one of them is 1. Note that any bicyclic graph is obtained from an ∞-graph or a θ -graph by attaching trees to some of its vertices. Similarly, for any tricyclic graph G, the base of G, denoted by GB , is the minimal tricyclic subgraph of G. Clearly GB is the unique tricyclic of G containing no pendant vertex, and G can be obtained from GB by attaching trees to some vertices of GB . It follows from [8] that there are eight types of bases for tricyclic graph, which are depicted in Fig. 2. In [5], Collatz and Sinogowitz first posed the problem of characterizing all graphs that satisfy η(G) > 0. This question is of great interest in chemistry, because η(G) = 0 is a necessary (but not sufficient) condition for a so-called conjugated molecule to be chemically stable, where G is the graph representing the carbon-atom skeleton of this molecule; The nullity of a graph is also important in mathematics, since it is related to the singularity of A(G). In recent years, this problem has been paid much attention by many researchers (see [3,4,9,11,12] and the references therein). It was shown in [15] that the nullity η(G) of G is bounded by an upper bound and a lower bound as
|V (G)| − 2m(G) − c(G) ≤ η(G) ≤ |V (G)| − 2m(G) + 2c(G). Graphs with nullity attaining the upper bound and the lower bound have been characterized by Song et al. [13] and Wang [14], respectively. In this paper, we prove that there is no graph with nullity η(G) = |V (G)| − 2m(G) + 2c(G) − 1.
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Furthermore, for fixed c(G), infinitely many connected graphs with nullity |V (G)|− 2m(G) + 2c(G) − k, (0 ≤ k ≤ 3c(G), k ̸ = 1) are also constructed. 2. Lemmas and results By the well known Cauchy-interlacing theorem, we have the following. Lemma 1 ([2]). If x is a vertex of a graph G, then η(G) − 1 ≤ η(G − x) ≤ η(G) + 1. Lemma 2 ([15]). For any graph G, |V (G)| − 2m(G) − c(G) ≤ η(G) ≤ |V (G)| − 2m(G) + 2c(G). Let G be a graph with pairwise vertex-disjoint cycles (if any), and let C (G) denote the set of all cycles in G. The acyclic graph TG is obtained from G by contracting cycles. More specifically, the vertex set V (TG ) is taken to be U ∪ WG , where U consists of all vertices of G that do not lie on a cycle and WG = {vC : C ∈ C (G)}. Two vertices in U are adjacent in TG if and only if they are adjacent in G, a vertex u ∈ U is adjacent to a vertex vC ∈ WG if and only if u is adjacent (in G) to a vertex in the cycle C . It is clear that TG is always acyclic and is a tree when G is connected. The following result determined all the graphs with nullity attaining the upper bound of Lemma 2. Lemma 3 ([13]). For any graph G, η(G) = |V (G)| − 2m(G) + 2c(G) if and only if the following three conditions are all satisfied: (a) Distinct cycles of G (if any) are vertex-disjoint; (b) The length of each cycle of G (if any) is a multiple of 4; (c) m(TG ) = m(TG − WG ) or, equivalently, there exists a maximum matching of TG that does not cover any vertex in WG . For convenience, hereafter in this paper we say a graph G satisfies the maximal nullity condition if η(G) = |V (G)| − 2m(G) + 2c(G). Definition 1. Let G be a graph with at least one pendant vertex. The operation of deleting a pendant vertex and its adjacent vertex from G is called PED (short form for pendant edge deletion). Lemma 4 ([6]). Let G be a graph and H be the graph obtained from G by PED operation. Then η(G) = η(H). Lemma 5 ([7]). For every acyclic graph T with at least one vertex, η(T ) = |V (T )| − 2m(T ). Lemma 6 ([7]). η(Pn ) equals 1 if n is odd and equals 0 if n is even; η(Cn ) equals 2 if n is a multiple of 4 and equals 0, otherwise. Lemma 7 ([10]). Suppose G is a unicyclic graph with n vertices and the cycle in G is Cl . Let E1 be the set of edges of G between Cl and G − Cl and E2 the set of matchings of m(G) edges. Then (1) η(G) = n − 2m(G) − 1 if m(G) = (l − 1)/2 + m(G − Cl ); (2) η(G) = n − 2m(G) + 2 if G satisfies properties: m(G) = l/2 + m(G − Cl ), l = 0(mod 4) and E1 ∩ M = ∅ for all M ∈ E2 ; (3) η(G) = n − 2m(G) otherwise. Remark: From Lemma 7, there exist unicyclic graphs with nullity n − 2m(G) − 1 = |V (G)|− 2m(G) − c(G), n − 2m(G) and n − 2m(G) + 2, respectively. The unicyclic graph does not exist only for nullity n − 2m(G) + 1 = |V (G)| − 2m(G) + 2c(G) − 1. An induced subgraph S of a graph G is called a pendant star of G if S is a star with at least two vertices such that the center of S is the only vertex which is adjacent to some vertices not in S. Lemma 8 ([10]). If S is a pendant star of a graph G, then m(G) = 1 + m(G − S). Furthermore, G has a maximum matching which consists of edges in a maximum matching for G − S together with an edge of S. From Lemma 8, we immediately have the following corollary: Corollary 1.
Let G be a graph with a pendant vertex u, and uv ∈ E(G). Then m(G) = m(G − u − v ) + 1.
Lemma 9 ([1]). A matching M of graph G is a maximum matching if and only if G contains no M-augmenting path.
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Lemma 10. Let G be a graph with a pendant vertex u, uv ∈ E(G) and v does not lie on any cycle of G. Then η(G) = |V (G)| − 2m(G) + 2c(G) − k, (0 ≤ k ≤ 3c(G)) if and only if η(G − u − v ) = |V (G − u − v )| − 2m(G − u − v ) + 2c(G − u − v ) − k. Proof. It is easy to get
|V (G)| = |V (G − u − v )| + 2.
(1)
|E(G)| = |E(G − u − v )| + d(v ).
(2)
and
θ (G − u − v ) = θ (G) + d(v ) − 2.
(3)
By Eqs. (1)–(3), we can get c(G) = |E(G)| − |V (G)| + θ (G)
= |E(G − u − v )| + d(v ) − |V (G − u − v )| − 2 + θ (G − u − v ) − d(v ) + 2 = |E(G − u − v )| − |V (G − u − v )| + θ (G − u − v ) = c(G − u − v ).
(4)
Then by Eqs. (1), (4), Corollary 1 and Lemma 4, we have
η(G − u − v ) = η(G) = |V (G)| − 2m(G) + 2c(G) − k = |V (G − u − v )| + 2 − 2(m(G − u − v ) + 1) + 2c(G − u − v ) − k = |V (G − u − v )| − 2m(G − u − v ) + 2c(G − u − v ) − k. □ Lemma 11. Let G be a graph with a pendant vertex u, and uv ∈ E(G). If v lies on some cycle of G, then η(G) ≤ |V (G)| − 2m(G) + 2c(G) − 2. Proof. We prove the result by contradiction. From Lemma 2, we can suppose that
η(G) = |V (G)| − 2m(G) + 2c(G) − k, k = 0, 1.
(5)
Note that v lies on some cycle of G, then
θ (G − u − v ) ≤ θ (G) + d(v ) − 3. Thus we have c(G) = |E(G)| − |V (G)| + θ (G)
≥ |E(G − u − v )| + d(v ) − |V (G − u − v )| − 2 + θ (G − u − v ) − d(v ) + 3 = |E(G − u − v )| − |V (G − u − v )| + θ (G − u − v ) + 1 = c(G − u − v ) + 1.
(6)
Then by Eqs. (5), (6), Corollary 1 and Lemma 4, we have
η(G − u − v ) = η(G) = |V (G)| − 2m(G) + 2c(G) − k ≥ |V (G − u − v )| + 2 − 2(m(G − u − v ) + 1) + 2(c(G − u − v ) + 1) − k = |V (G − u − v )| − 2m(G − u − v ) + 2c(G − u − v ) + 2 − k (k = 0, 1), a contradiction to Lemma 2.
□
Let F be the graph in Fig. 3. For any vertex v of graph F , it is easy to see that F −v is either K1,3 or C4 . Let D = F ∪(n−5)K1 . Then by Lemmas 3 and 5, we have η(D) ̸ = |V (D)| − 2m(D) + 2c(D) but η(D − v ) = |V (D − v )| − 2m(D − v ) + 2c(D − v ). Lemma 12. Let G ̸ = D be a graph without pendant vertices. If η(G) ̸ = |V (G)| − 2m(G) + 2c(G) and c(G) ≥ 2, then there exists a vertex, say x, on some cycle of G such that η(G − x) ̸ = |V (G − x)| − 2m(G − x) + 2c(G − x). Proof. If g(G) = 3, and c(G) ≥ 2, then there must exist a vertex, say x, on some cycle of G such that G − x contains C3 as an induced subgraph. From Lemma 3, we have η(G − x) ̸ = |V (G − x)| − 2m(G − x) + 2c(G − x). In the following we suppose that g(G) ≥ 4. Since η(G) ̸ = |V (G)| − 2m(G) + 2c(G), G does not satisfy at least one of the three cases in Lemma 3. Without loss of generality, we distinguish the following three cases.
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Fig. 3. Graph F .
Case 1. Suppose that G does not satisfy case (a) of Lemma 3. Then there exist at least two vertex-joint cycles, say Ck (k ≥ 4) and Cs (s ≥ 4), in G. Let G(Ck , Cs ) be the subgraph induced by Ck and Cs . The following two subcases can be identified for this case. Subcase 1.1. c(G) = 2. Then G is the union of G(Ck , Cs ) and some isolated vertices (if any), and G(Ck , Cs ) is either ∞-graph (l = 1) or θ -graph (see Fig. 1). Since G ̸= D, there exists a vertex, say x, on some cycle of G such that G − x has a cycle with a pendant edge. By Lemma 11, we have η(G − x) ̸ = |V (G − x)| − 2m(G − x) + 2c(G − x). Subcase 1.2. c(G) ≥ 3. If there exists a vertex x on some cycle of G and x ∈ / G(Ck , Cs ), then η(G − x) ̸= |V (G − x)| − 2m(G − x) + 2c(G − x) by Lemma 3; otherwise, G(Ck , Cs ) contains one of tricyclic graphs e, f , g , h (see Fig. 2) as a subgraph. Then there exists a vertex, say x on some cycle of G such that G − x has two vertex-joint cycles. Then from Lemma 3, we have η(G − x) ̸ = |V (G − x)| − 2m(G − x) + 2c(G − x). Case 2. Suppose G does not satisfy case (b) but satisfies case (a) of Lemma 3. Then there exists at least one cycle with length is not a multiple of 4. For c(G) ≥ 2, there exists a vertex x on another cycle of G such that η(G − x) ̸ = |V (G − x)| − 2m(G − x) + 2c(G − x) by Lemma 3. Case 3. Suppose G does not satisfy case (c) but satisfies cases (a) and (b) of Lemma 3. If E(TG ) = ∅, then G is the union of some cycles and some isolated vertices. So m(TG ) = m(TG − WG ) = 0, a contradiction to the assumption. Suppose E(TG ) ̸ = ∅ in the following. For every maximum matching M of TG , M must contain at least one pendent edge of TG . Otherwise we can find an M-augmenting path in TG , a contradiction with Lemma 9. Let u be a pendent vertex of TG . Since G has no pendant vertex, u ∈ WG . It is easy to see that the cycle in G corresponding to u is a pendant cycle, denoted by Cu . Let v be the unique vertex of Cu adjacent to a vertex outside Cu and x (x ̸ = v ) be a vertex of Cu . Then TG−x is the graph obtained from TG and Cu − x by identifying u and v as one vertex. The following two subcases can be identified for this case. Subcase 3.1. Suppose that every maximum matching of TG must contain all pendant edges of TG . Let x be one of two vertices of Cu such that x is adjacent to v . Note that the length of Cu is a multiple of 4. Then m(TG−x ) = m(TG ) + m(Cu − x) and M(TG ) ∪ M(Cu − x − v ) is a maximum matching of TG−x , where M(TG ) and M(Cu − x − v ) are the maximum matchings of TG and Cu − x − v , respectively. It is easy to see that M(TG ) ∪ M(Cu − x − v ) covers some vertex in WG−x . Then η(G − x) ̸= |V (G − x)| − 2m(G − x) + 2c(G − x) by Lemma 3. Subcase 3.2. Otherwise. There exist some pendant edge, say w u, and some maximum matching, say M(TG ), of TG such that w u ∈ / M(TG ). Let y be a vertex of Cu such that d(v, y) = 2. Then m(TG−y ) = m(TG ) + m(Cu − y) and M(TG ) ∪ M(Cu − y) is a maximum matching of TG−y , where M(Cu − y) is the maximum matching of Cu − y. It is easy to see that M(TG ) ∪ M(Cu − y) covers some vertex in WG−y . Then η(G − y) ̸ = |V (G − y)| − 2m(G − y) + 2c(G − y) by Lemma 3. □ Theorem 2.1.
Let G be a graph without pendant vertices. Then we have
η(G) ̸= |V (G)| − 2m(G) + 2c(G) − 1. Proof. If G = D, it is easy to see that η(G) ̸ = |V (G)| − 2m(G) + 2c(G) − 1. Suppose that G ̸ = D in the following. We prove the result by induction on c(G). If c(G) = 0, then G = nK1 , the result is obvious. If c(G) = 1, then G = Ck ∪ (n − k)K1 , (3 ≤ k ≤ n). By Lemma 6, the result follows. Now suppose that the result holds for c(G) ≤ k, (k ≥ 1). We next need to prove that the result holds for c(G) = k + 1. Suppose there exists some graph G with c(G) = k + 1, and
η(G) = |V (G)| − 2m(G) + 2c(G) − 1.
(7)
For the graph G − x, x is any vertex on the cycle of G, we have m(G) ≤ m(G − x) + 1.
(8)
θ (G − x) ≤ θ (G) + d(x) − 2.
(9)
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Fig. 4. G(k1 , k2 , k3 ).
By Eq. (9), we can get c(G) = |E(G)| − |V (G)| + θ (G)
≥ |E(G − x)| + d(x) − (|V (G − x)| + 1) + θ (G − x) − d(x) + 2 = c(G − x) + 1.
(10)
Then by Eqs. (7), (8) and (10), we have
η(G) = |V (G)| − 2m(G) + 2c(G) − 1 ≥ |V (G − x)| + 1 − 2(m(G − x) + 1) + 2(c(G − x) + 1) − 1 = |V (G − x)| − 2m(G − x) + 2c(G − x).
(11)
By Eq. (10), c(G − x) ≤ k, where x is any vertex on the cycle of G. Then by the assumption for induction, we have η(G − x) ̸= |V (G − x)| − 2m(G − x) + 2c(G − x) − 1. From Lemma 12, there exists some vertex, say y on the cycle of G, such that η(G − y) ̸ = |V (G − y)| − 2m(G − y) + 2c(G − y). So we have η(G − y) ≤ |V (G − y)| − 2m(G − y) + 2c(G − y) − 2. Then by Lemma 1, we have η(G) ≤ η(G − y) + 1 ≤ |V (G − y)| − 2m(G − y) + 2c(G − y) − 1, a contradiction to Eq. (11). □ Now we give the main result of this paper. Theorem 2.2. For any graph G, η(G) ̸ = |V (G)| − 2m(G) + 2c(G) − 1. Proof. If G is an acyclic graph, by Lemma 5, the result follows. In the following, we suppose that G is not an acyclic graph. We prove the result by contradiction. Let G be not an acyclic graph with nullity η(G) = |V (G)| − 2m(G) + 2c(G) − 1. If G has a pendant edge, say uv ∈ E(G), where u is a pendant vertex and v does not lie on any cycle of G. By applying PED operation to G, we have η(G − u − v ) = |V (G − u − v )| − 2m(G − u − v ) + 2c(G − u − v ) − 1 from Lemma 10. Going on the above process at last we can get a graph H such that either H has no pendant vertices or H has a pendant edge, say wr, where r is a pendant vertex and w lies on some cycle of H, and η(H) = |V (H)| − 2m(H) + 2c(H) − 1, a contradiction to Theorem 2.1 or Lemma 11. □ Let G(k1 , k2 , k3 ) (ki ≥ 0, i = 1, 2, 3) be the connected graph obtained from k1 C3 , (k2 + k3 )C4 and K1,k3 +1 by adding k1 + k2 + k3 new edges among them (see Fig. 4). Theorem 2.3. For fixed c(G), there are infinitely many connected graphs with the nullity |V (G)| − 2m(G) + 2c(G) − k, (0 ≤ k ≤ 3c(G), k ̸ = 1). Proof. Let G(k1 , k2 , k3 ) (k1 + k2 + k3 = c(G(k1 , k2 , k3 ))) be the graph defined as Fig. 4. It is easy to see that
|V (G(k1 , k2 , k3 ))| = 3k1 + 4k2 + 5k3 + 2; m(G(k1 , k2 , k3 )) = k1 + 2k2 + 2k3 + 1; c(G(k1 , k2 , k3 )) = k1 + k2 + k3 .
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Since η(K1,2 ) = 1, η(C4 ) = 2, η(C3 ) = 0, from Lemma 4 and the above three equations, we have
η(G(k1 , k2 , k3 )) = η(G(k1 , k2 , k3 ) − u − v ) (by Lemma 4) = 2k2 + k3 = |V (G(k1 , k2 , k3 ))| − 2m(G(k1 , k2 , k3 )) + 2c(G(k1 , k2 , k3 )) − 3k1 − 2k3 . It is easy to see that 3k1 + 2k3 can take over every integer from zero to 3c(G) except for one. From Lemma 10, the result follows. □ References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16]
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