Automorphism Group and Other Properties of Zero Component Graph over a Vector Space

In this paper, we introduce an undirected simple graph, called the zero component graph on finite-dimensional vector spaces. It is shown that two finite-dimensional vector spaces are isomorphic if and only if their zero component graphs are isomorphic, and any automorphism of a zero component graph can be uniquely decomposed into the product of a permutation automorphism and a regular automorphism. Moreover, we find the dominating number, as well as the independent number, and characterize the minimum independent dominating sets, maximum independent sets, and planarity of the graph. In the case that base fields are finite, we calculate the fixing number and metric dimension of the zero component graphs and determine vector spaces whose zero component graphs are Hamiltonian.


Introduction
A binary relation on a vector space, group, or ring can be studied with the help of considering the associating graph defined by this relation. Symplectic graphs, orthogonal graphs, subspace inclusion graphs, and nonzero component graphs are such examples which have been considered recently (see [1][2][3][4]).
In this paper, we always assume n ≥ 2, F is a field, F * � F\ 0 { }, and V is an n-dimensional vector space over F. Let A � ε 1 , . . . , ε n be a basis of V. en, any α ∈ V can be uniquely written as α � n i�1 x i ε i with x i ∈ F, and S A (α) (resp., Z A (α)) is the set of all basis vectors whose coefficients are nonzero (resp., zero) in the linear expression of α with respect to (i.e., with respect to) A, namely, Clearly, S A (α) ∪ Z A (α) � A and S A (α) ∩ Z A (α) � ∅. All graphs considered in this paper are undirected simple graphs (without loops and multiedges). e nonzero component graph Γ c (V) of V, introduced by Das [1], is a graph with vertex set V * � V\ 0 { }, in which for distinct α, β ∈ V * , there exists an edge joining α and β if and only if S A (α) ∩ S A (β) ≠ ∅. In [1], the author studied the diameter, dominating number, and dependent number of Γ c (V). When the base field F is finite, Das [5] considered the clique number, edge connectivity, and chromatic number of Γ c (V) and showed that Γ c (V) is Hamiltonian, but not Eulerian. Murtaza et al. [6] considered two parameters, called the locating-dominating number and identifying number, of Γ c (V). Also, the nonzero component union graph Γ u (V) of V has V * as its vertex set, but two vertices α, β ∈ V * are adjacent if and only if〈S A (α), S A (β)〉 � V, where 〈S〉 stands for the subspace of V generated by a subset S⊆V. In [7], Das studied some properties of Γ u (V), such as the connectedness, dominating number as well as maximal cliques and showed that Γ u (V) is weakly perfect when the base field F is finite.
It is well known that the automorphism group of a graph reflects the symmetry of the graph. Generally, it is an important but difficult work to describe the full automorphism group both in graph theory and in algebra. In [1], the author characterizes the automorphisms of Γ c (V). For the automorphisms of symplectic graphs, orthogonal graphs, and subspace inclusion graphs, the reader is referred to [3,4,8]. e fixing number and metric dimension of a graph (see Section 2 for the definitions) are two parameters which "destroy" the automorphisms and symmetry. Moreover, the fixing number has been used to consider the problem of programming a robot to manipulate objects (see [9]), while the metric dimension has been applied to various areas, such as pharmaceutical chemistry, robot navigation, diverse as combinatorial optimization, and sonar (see [10]). Recently, several authors considered the fixing number and metric dimension of some associating graphs defined on a vector space. Fazil [11] computed the fixing number of Γ c (V), and Benish [12] studied the relation between fixing number and another parameter, the fixed number, of Γ c (V). Ali et al. [13] calculated the metric dimension of Γ c (V). For a positive integer k < n, the Kneser graph K(n, k) is a graph with all k-subsets of [n] � 1, . . . , n { } as vertex set, in which two distinct k-subsets are connected by an edge if they are disjoint. Also, the vertex set of the Johnson graph J(n, k) consists of all k-subsets of [n], but two distinct k-subsets are adjacent if their intersection is a (k − 1)-subset. In [14], Boutin obtained sharp bounds for the fixing number of Kneser graphs and determined all Kneser graphs whose fixing number is 2, 3, or 4. In [15], using various algebraic, combinatorial, and geometric approaches, Bailey et al. studied the constructions of resolving sets of Kneser and Johnson graphs and provided bounds on their metric dimension. For more works on the fixing number and metric dimension, the reader is referred to [9,[16][17][18][19][20][21] and the references therein.
If A is a basis of V, then it is clear that V * can be partitioned into n classes: On the zero component graph Γ(V, A) of V with respect to A, we mean it is a graph defined as follows: Motivated by all the above, we intend to investigate the properties of Γ(V, A). e rest of this paper is structured as follows: in Section 2, we introduce some definitions and notation in graph theory. In Section 3, we show that the zero component graph does not depend on the choice of the bases and thus restate the definition of the zero component graph (see Definition 1). In Section 4, we prove that two finite-dimensional vector spaces are isomorphic if and only if their zero component graphs are isomorphic. e automorphisms of the zero component graph are determined in Section 5, and the fixing number and metric dimension are computed in Section 6. Finally, we consider other properties of the zero component graph, such as the girth, minimum independent dominating sets, maximum independent sets, planarity, and hamiltonicity in Section 7.

Notation and Preliminaries
Let G be a graph with a vertex set V(G). Two vertices x, y ∈ V(G) are said to be adjacent and written as x ∼ y if there exists an edge joining x and y, otherwise, nonadjacent and u≁v. A graph is said to be complete if any pair of its vertices are adjacent; as usual, K n stands for the complete graph of order n. Let N(x) (resp., deg(x)) be the neighbors (resp., degree) of x, that is, then, x and y are said to be twins. We use Tw(x) to denote the twin set of x, which consists of all twins of x in G. In particular, a vertex x ∈ V(G) is called a singleton whenever Tw(x) � x { }. Clearly, the relation " ≈ " is equivalent if it is defined as x ≈ y⇔x ∈ Tw(y); and then V(G) is a union of some disjoint twin sets of G. e number of twin sets of G is written as tw(G).
For S⊆V(G), it is called a dominating set of G if any vertex in V(G)\S is adjacent to at least one vertex in S, and it is called an independent set of G if any pair vertices in S are nonadjacent in G. Also, a dominating set is called an independent dominating set if it is an independent set. e minimum dominating set (resp., maximum independent set, minimum independent dominating set) of G is a dominating set (resp., independent set, independent dominating set) with the minimum (resp., maximum, minimum) cardinality, and the cardinality of a minimum dominating set (resp., maximum independent set) is called the dominating number (resp., independent number) of G.
A trail in G is an alternating sequence of vertices and edges, x 0 ∼ x 1 ∼ · · · ∼ x k for some positive integer k, where x i 's are not necessarily distinct; it is called a path if x i 's are distinct except for possibly the first and last vertices; and this path (resp., trail) is said to be joining x 0 and x k . A graph is called connected if, for any pair of vertices, there exists a path joining them; it is called k-connected if each induced subgraph obtained by the graph from deleting k − 1 vertices is connected. e connectivity of G is the maximum value of k for which G is k-connected. A cycle of length k is a path x 0 ∼ x 1 ∼ · · · ∼ x k with x 0 � x k ; also, a cycle of length 3 is called a triangle. e girth of G is the minimum of the lengths of all cycles in G. Moreover, a graph is called Eulerian (resp., Hamiltonian) if there exists a trail (resp., cycle) which contains all edges (resp., all vertices) in it exactly once.
For graphs G and H, they are called isomorphic if there exists an isomorphism ϕ from G to H, that is, An automorphism of G is an isomorphism from G to itself. e set Aut(G) of all automorphisms of G forms a group under the composition of the transformations. G is said to be vertex transitive if, for any x, y ∈ V(G), there exists ϕ ∈ Aut(G) such that ϕ(x) � y. A subset W⊆V(G) is called a fixing set (or determining set) of G if the only automorphism of G that fixes every vertex in W is the identity, and the fixing number (or determining number) of G, denoted by fix(G), is the smallest size of such a set. For x, y ∈ V(G), the distance between them, written as d(x, y), is the length of the shortest path joining them, and the diameter of G is the largest distance between pairs of vertices of G. If W � w 1 , w 2 , . . . , w k is an ordered subset of V(G), then the k-dimensional vector (d(x, w 1 ), . . . , d (x, w k )) is called the representation of x with respect to W, and x is said to be resolved by W if (d(x, w 1 ), . . . , d(x, w k )) ≠ (d(y, w 1 ), . . . , d(y, w k )) for any other vertex y ∈ V(G). Moreover, W is called a resolving set (or locating set) of G if distinct vertices of G have distinct representations with respect to W, and the metric dimension (or locating number) of G, denoted by dim(G), is the minimum cardinality of such a set. A graph G is called an FED-graph if fix(G) � dim(G). All other unexplained notations and definitions on graph theory are standard (see [22] or [23] for details).
Here, we list some results on graphs, which will be used in this paper.
e number of edges in a graph G is Proposition 2 (see [22], p. 18). A graph as well as its complement has the same automorphisms.
Proposition 3 (see [11], eorem 1). A finite graph and its complement are of the same fixing number.

5). A connected graph is Eulerian if and only if each vertex is of even degree.
Proposition 6 (Kuratowski's theorem, [22], eorem 10.30). A graph is planar if and only if it contains no subdivisions of K 5 or K 3,3 , where K n,n stands for the balanced complete bipartite graph of order 2n.
Proposition 7 (see [22], Corollary 18.12). If G is a connected graph in which all vertices are of odd degree, then it is Hamiltonian.
Proposition 9 (see [25]). Let G be a 2-connected graph with minimum degree δ and independent number ι.

Zero Component Graph of a Vector Space
We first show that the zero component graph of V does not depend on the choice of the bases. Proof. Clearly, there exists an invertible linear transfor- In case the zero component graph of V does not depend on the choice of the bases, two vectors may be adjacent with respect to one basis but not adjacent with respect to another basis. For example, suppose that V is a 3dimensional vector space over F, As usual, denote by e i ∈ V, 1 ≤ i ≤ n, the vector with the i-th component 1 and the others 0. It is well known that E � e 1 , . . . , e n constructs a basis of V. eorem 1 shows that we need only to consider the zero component graph For the convenience of writing, we set e � n i�1 e i and use en, by eorem 1, we can restate the definition of the zero component graph as follows.
Next, we describe the connectedness and diameter of Γ(V).

Zero Component Graph and Graph Isomorphism Problem
In this section, we investigate the interrelationship between the isomorphism of two vector spaces and that of their zero component graphs. Before that, we study the twins and the number of twin sets of the zero component graph.
For (ii), we see that the case of |F| � 2 is clear according to Lemma 2. Assume |F| ≠ 2. For any α ∈ V * and any x ∈ F\ 0, 1 { }, it is obvious that α and xα are twins. is shows that α is not a singleton in Γ(V), which confirms (ii). e following result is a direct consequence of Lemma 1 (i). In the following, we will prove that two vector spaces over distinct finite fields are isomorphic if and only if their zero component graphs are isomorphic. To this end, we give the vertex degree of Γ(V).
Clearly, any pair of N 1 (β), . . . , N j (β) do not intersect with each other, and For any α ∈ Z j , we write α � n− j i�1 a i e k i with a i ∈ F * and 1 ≤ k 1 < · · · < k n− j ≤ n. Take a permutation σ on [n] such that σ(i) � k i for 1 ≤ i ≤ j, and define the map ϕ σ : V ⟶ V by ϕ σ ( n i�1 y i e i ) � n i�1 y i e σ(i) for any y i ∈ F. en, the restriction of ϕ σ on V * is an automorphism of Γ(V), and By Lemma 4, we get the following results immediately. □ Corollary 1. Let n ≥ 3, and q � |F|. en, the order and size m of Proof. It is easily seen that the order of Γ(V) is |V * | � |V| − |Z 0 | − 1 � q n − (q − 1) n − 1. For the size of Γ(V), by Handshaking lemma (see Proposition 1), Lemma 4, from which it follows the result.

Corollary 3. Let n ≥ 3, and let F be a finite field. en, Γ(V) is Eulerian if and only if |F| is even.
Proof. When |F| is even (resp., odd), by Lemma 4, we obtain that each vertex in Γ(V) is of even degree (resp., odd degree), which shows Γ(V) is (resp., not) Eulerian. Finally, we show that two vector spaces over distinct finite fields are isomorphic if and only if their zero component graphs are isomorphic. □ Theorem 3. Let n, m ≥ 2; V an n-dimensional vector space over a field with q 1 elements; and W an m-dimensional vector space over a field with q 2 elements. en, V is isomorphic to W

as vector spaces if and only if Γ(V) is isomorphic to Γ(W).
Proof. e necessity is obvious; thus, it suffices to prove the sufficiency. Assume that Γ(V) is isomorphic to Γ(W). If n � 2, then by Lemma 1 (i), we know that Γ(V) is a union of two complete graphs each of which is of order q 1 − 1, and so is Γ(W). Again, by Lemma 1 (i), we have m � 2 and q 1 � q 2 . Similarly, one may get n � 2 and q 1 � q 2 when m � 2. Hence, V and W are isomorphic as vector spaces when n � 2 or m � 2. Now, let n, m ≥ 3. Lemma 3 (i) shows that the numbers of twin sets of Γ(V) and Γ(W) are 2 n − 2 and 2 m − 2, respectively, which follows n � m. On the other hand, the minimum degrees of Γ(V) and Γ(W) are, respectively, q n− 1 1 − 2 and q m− 1 2 − 2 according to Corollary 2. en, by n � m, we obtain q 1 � q 2 .
us, V and W are isomorphic as vector spaces.

Automorphism of Γ(V)
We first construct two types of standard automorphisms of Γ(V) as follows.

Definition 2. Let G be a graph and ρ a bijection on V(G).
If ρ permutates the twins in G, then it is called a regular automorphism of G.

Lemma 5 (i) Let ρ and ϕ σ be defined as above. en, ϕ − 1 σ · ρ · ϕ σ is a regular automorphism of Γ(V). (ii) If an automorphism ϕ of Γ(V) is a regular automorphism but also a permutation automorphism, then ϕ is the identity.
Proof. For any α � n i�1 a i e i ∈ V * with a i ∈ F, it is clear that ρϕ σ (α) � ρ( n i�1 a i e σ(i) ) and n i�1 a i e σ(i) are twins. By Lemma 2, we assume ρ( n i�1 a i e σ(i) ) � n i�1 a i b i e σ(i) with b i ∈ F * . en, Again, by Lemma 2, we know that ϕ − 1 σ ρϕ σ (α) and α are twins. By the arbitrariness of α, we prove (i).
Next, we need to present a result on the zero-divisor graph. Let R be a commutative ring, then the zero-divisor graph Γ z (R) of R is a graph defined as follows: Γ z (R) has vertex set consisting of all nonzero zero-divisors of R, and two nonzero zerodivisors are adjacent in Γ z (R) if and only if their product in R is 0. If |F| � 2, we denote by Z n 2 the ring which has V as an additive group and the multiplication of two elements α � n i�1 a i e i , β � n i�1 b i e i ∈ V with a i , b i ∈ F are defined naturally: It is clear that V * is also the vertex set of Γ z (Z n 2 ). In [24], the automorphisms of Γ z (Z n Proposition 10 (see [24], eorem 1). Any automorphism of Γ z (Z n 2 ) is a permutation automorphism, and Aut(Γ z (Z n 2 )) � Sym(n). e following result shows that the relation between Γ(V), for |F| � 2, and Γ z (Z n 2 ).

Theorem 4. Let n ≥ 2. en, a bijection on V * is an automorphism of Γ(V) if and only if it can be uniquely decomposed into the product of a permutation automorphism and a regular automorphism of Γ(V).
Proof.
e uniqueness of decomposition is clear according to Lemma 5 (ii); thus, it suffices to prove the existence. When |F| � 2, we get the result by Lemma 7. Now, assume |F| ≠ 2.

Fixing Number and Metric Dimension of Γ(V)
In this section, we always assume F is a finite field with q( ≥ 2) elements. If n � 2, nothing needs to do for the metric dimension of Γ(V) according to Lemma 2; the fixing number Γ(V) is 1 when q � 2, and 2(q − 2) otherwise. In the following, we consider the case for n ≥ 3. Firstly, by Proposition 4, Lemma 3, and Corollary 1, we have the following result immediately. Lemma 9. Let n ≥ 3 and q ≥ 3. en, Γ(V) is an FED-graph, and fix(Γ(V)) � dim(Γ(V)) � q n − (q − 1) n − 2 n + 1.
Next, we give the value of the fixing number of Γ(V) for n ≥ 3 and q � 2.
□ Lemma 11. If q � 2, then all vectors in V * are of different types.
Now, we can calculate the metric dimension of Γ(V) for n ≥ 3 and q � 2.
e case for n � 2 is clear according to Lemma 1 (i).
is is a direct consequence of the fact that any vertex is adjacent to at least one of e 1 , e 2 . □ Theorem 7. Let n ≥ 3. en, I � α, β ⊆V * is a minimum independent dominating set of Γ(V) if and only if Z(α) ∩ Z(β) � ∅ as well as Z(α) ∪ Z(β) � e 1 , . . . , e n .

Lemma 14.
e independence number of Γ(V) is n (i.e., the dimension of V ).
Proof. e sufficiency is obvious; thus, we need only to prove the necessity. Let I � α j |1 ≤ j ≤ n ⊆V * be an independent set of Γ(V). Clearly, Z(α 1 ), . . . , Z(α n ) are disjoint. It suffices to show α j ∈ Z 1 ; that is, α j has exactly one zero component, 1 ≤ j ≤ n. e case for n � 2 is clear. Let n ≥ 3, and suppose, to the contrary, there exists some 1 ≤ k ≤ n such that α k ∉ Z 1 . Without loss of the generalization, assume e n− 1 , e n ∈ Z(α k ). For any j ≠ k, by α j ≁α k , we get e n− 1 , e n ∉ Z(α j ), and consequently ∪ j≠k Z(α j )⊆ e i |1 ≤ i ≤ n − 2}.
Next, we provide a lower bound for the connectivity of Γ(V) when F is a finite field. □ Lemma 15. Let n ≥ 3 and q � |F|. en, the connectivity of Γ(V) is greater than or equal to q n− 2 − 1.
Finally, we can characterize the hamiltonicity of Γ(V) when F is a finite field.  Proof. If |F| is odd, then each vertex in Γ(V) is of odd degree (see Lemma 4). By Proposition 7, we know that Γ(V) is Hamiltonian. Now, let |F| be even. If n ≥ 5 or |F| ≥ 4, we have |F| n− 2 − 1 ≥ n; that is, the connectivity of Γ(V) is greater than or equal to the independent number (see Lemma 14 and Lemma 15); then, by Chvátal-Erdős theorem (see Proposition 8), we obtain that Γ(V) is Hamiltonian. e case for n � 3 and |F| � 2 is clear according to Figure 1. If n � 4 and |F| � 2, then Γ(V) is 2-connected with minimum degree δ � 2 n− 1 − 2 (see Corollary 2) and independent number n, which follows δ ≥ max (|V * | + 2/3), n . Applying a theorem of Nash-Williams (see Proposition 9), we see that Γ(V) is Hamiltonian.

Conclusion
In this paper, we present an undirected simple graph, called the zero component graph Γ(V), over a finite-dimensional vector space V, and investigate various interrelationships among Γ(V) and V. e aim of this paper is to determine the automorphisms, compute the fixing number as well as metric dimension, and study other properties such as girth, minimum independent dominating sets, maximum independent sets, planarity, and hamiltonicity of Γ(V). In addition, a lower bound for the connectivity is given. As a topic of further research, one can give the exact value of the connectivity and clique number of Γ(V).

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare no conflicts of interest.