Abstract
Resolving parameters are a fundamental area of combinatorics with applications not only to many branches of combinatorics but also to other sciences. In this study, we construct a class of Toeplitz graphs and will be denoted by so that they are Cayley graphs. First, we review some of the features of this class of graphs. In fact, this class of graphs is vertex transitive, and by calculating the spectrum of the adjacency matrix related with them, we show that this class of graphs cannot be edge transitive. Moreover, we show that this class of graphs cannot be distance regular, and because of the difficulty of the computing resolving parameters of a class of graphs which are not distance regular, we regard this as justification for our focus on some resolving parameters. In particular, we determine the minimal resolving set, doubly resolving set, and strong metric dimension for this class of graphs.
1. Introduction
The graphs in this paper are simple, undirected, and connected. An automorphism of a graph is a permutation of the vertex set of with the property that, for any vertices and , we have is adjacent to in if and only if is adjacent to in . The set of all automorphisms of a graph , with the operation of composition of permutations, is a permutation group on and a subgroup of the symmetric group on . This is the automorphism group of , denoted by . Suppose and are two graphs. If there is a bijection, say , from to so that is adjacent to in if and only if is adjacent to in , then we say that is isomorphic to . If we consider a graph as a network, then the network stability is very important to us, and especially, if graph is vertex transitive, that is, acts transitively on , then the cost of studying the network will be very low, and hence, the network will be more stable. Consider a finite group , and suppose is a subset of so that it is closed under taking inverses and does not contain the identity; then, the Cayley graph has vertex set and edge set . Thus, the studying of Cayley graphs is very useful because every Cayley graph is vertex transitive [1]. The distance between any pair of vertices of is the length of geodesic between and , denoted by or simply . A vertex is said to resolve a pair if . Resolving parameters are a fundamental area of combinatorics with applications not only to many branches of combinatorics but also to other sciences. For an arranged subset of vertices in a connected graph , the metric representation of a vertex in is the -vector relative to . Also, the subset is considered as the resolving set for if any pair of vertices of is distinguished by some vertices of . A resolving set with least number of vertices is referred as metric basis for and the cardinality of such resolving set is known as metric dimension denoted by . The metric dimension of a graph is the least number of vertices in a set with the property that the list of distances from any vertex to those in the set uniquely identifies that vertex. The concept of the metric dimension in algebraic graph theory dates back to the 1970s. It was defined independently by Harary and Melter [2] and by Slater [3]. In recent years, a considerable literature has developed [4]. This concept has different applications in the areas of network discovery and verification [5]. For more details, see [6–9]. An (n x n) matrix is called a Toeplitz matrix if for each , see [10]. In fact, a Toeplitz matrix is a square matrix so that entries in every diagonal parallel to the main diagonal are equal, and hence, a Toeplitz matrix is determined by its first row and column. A simple undirected graph with vertex set and its adjacency matrix is called a Toeplitz graph if is the Toeplitz matrix. In this paper, we consider a class of Toeplitz graphs will be denoted by so that they are Cayley graphs as follows.
Let be a fixed even integer is greater than or equal ; also, let and be corresponding sets so that . Hence, we say that if . Now, let and be subsets of the set , and let be a refinement of union of the two sets and so that . We can see that a graph with vertices so that the vertices are labelled by the set and the edge set:which is Toeplitz graph . For more result of the Toeplitz graphs, see [11, 12]. Figure 1 shows the Toeplitz graph .
In particular, we can verify that the Toeplitz graph which is defined already is isomorphic to the Cayley graph , whereis the dihedral group of order , and is an inverse closed subset of . Thus, the Toeplitz graph is a vertex transitive. Also, for convenience, we can use the symbols in the Cayley graph , instead of the symbols in the Toeplitz graph . Some metrics for a class of distance regular graphs is computed in [13, 14]. On the contrary, because of the difficulty of the computing resolving parameters of a class of graphs which are not distance regular, we regard this as justification for our focus on some resolving parameters in the Cayley graph . The important results of this study are presented in Sections 3.1 and 3.2. In Section 3.1, first, we will be determining the automorphism group of the Cayley graph ; also, we will show that the Cayley graph cannot be distance regular. In particular, we will prove that the Cayley graph cannot be edge transitive. Moreover, in Section 3.2, we will be computing some resolving parameters for this class of Cayley graphs.
2. Definitions and Preliminaries
Definition 1 (see [15]). A graph is edge transitive if its automorphism group acts transitively on .
Definition 2 (see [15]). A graphis 1-transitive or symmetric if its automorphism group acts transitively on the set of paths of length 1 or 1-arcs.
Proposition 1 (see [15]). Letbe a symmetric graph of valency, and letbe a simple eigenvalue of; then,.
Definition 3 (see [16]). Suppose that is a regular graph of valencyand for any two verticesandin; if, then we have, and. Then, we say thatis a distance regular graph.
Proposition 2 (see [16]). Ifis a distance regular graph with diameter, thenhas exactlydistinct eigenvalues.
Definition 4 (see [17]). Supposeis a graph of order at least 2; verticesare said to doubly resolve verticesif. A setof vertices ofis a doubly resolving set ofif every two distinct vertices ofare doubly resolved by some two vertices of. A doubly resolving set with minimum cardinality is called the minimal doubly resolving set. This minimum cardinality is denoted by.
Definition 5 (see [18]). Letbe a graph. A vertexofstrongly resolves two verticesandofifbelongs to a shortestpath orbelongs to a shortestpath. A setof vertices ofis a strong resolving set ofif every two distinct vertices ofare strongly resolved by some vertex of. The strong metric dimension of a graphis the cardinality of smallest strong resolving set ofand denoted by.
3. Main Results
3.1. Some of the Features of the Cayley Graph
In this section, we review some of the features of the Cayley graph . It is well known that the spectrum of a graph is the spectrum of the adjacency matrix related with it, that is, its set of eigenvalues together with their multiplicities. If all the eigenvalues of the adjacency matrix of a graph are integers, in this case, the graph related with it is called an integral graph, see [19]. As we shall see, the theory of integral graphs has connections to some parts of graph theory, edge transitivity, and symmetric graph. In the next theorem, we obtain the automorphism group of the Cayley graph by applications of wreath product in graph theory; for more details of the wreath product, see [20].
Proposition 3. Letbe an even integer greater than or equal, andbe a Cayley graph on the dihedral group, whereis defined already. If, then, whereand
Proof. We can see that the complement of , denoted by , is isomorphic to the disjoint union of copies of cocktail party graph , and we can show that is isomorphic to the , where is the cyclic group of order and , see Proposition 3.2 of [21]. Hence, given by the above discussion and the theorem in [22], we have . In particular, we have because a simple undirected graph and its complement have the same automorphism group.
Proposition 4. Letbe an even integer greater than or equal, andbe a Cayley graph on the dihedral group, whereis defined already; then,cannot be a distance regular graph.
Proof. It is not hard to see that the diameter of is and is not a bipartite graph, because . Now, by a similar way, which is done in proof of Proposition 11 in [23], we can show that the adjacency matrix spectrum of is , where the superscripts give the multiplicities of eigenvalues with multiplicity greater than one. Hence, has exactly four distinct eigenvalues. Moreover, based on Proposition 2, we know that if is a distance regular graph with diameter , then has exactly distinct eigenvalues. Thus, cannot be a distance regular graph.
Proposition 5. Letbe an even integer greater than or equalandbe a Cayley graph on the dihedral group, whereis defined already; then,cannot be an edge transitive graph.
Proof. By contradiction, suppose is an edge transitive graph. It is well known that a connected graph that is edge transitive and vertex transitive need not be 1-transitive. In particular, in p.59, 7.53 of [24], Tutte proved that if a connected graph, regular of odd valency, is both vertex and edge transitive, then it is 1-transitive. Thus, if is a edge transitive graph, then it must be is a 1-transitive graph because it is vertex transitive of odd valency . On the contrary, based on Proposition 1, if is a simple eigenvalue of a 1-transitive graph , then , which is not the case, see Proposition 4. This contradiction shows that cannot be an edge transitive graph.
3.2. Metric Dimension, Minimal Doubly Resolving set, and Strong Resolving Set of the Cayley Graph
Theorem 1. Ifis an even integer greater than or equalandis a Cayley graph on the dihedral group, whereis defined already, then the metric dimension ofis.
Proof. Let , where and . For every pair of distinct vertices , the length of a shortest path from to is or because the diameter of is . In particular, if is an arranged subset of or in graph such that , then we can show that is not a resolving set of . Let be an arranged subset of vertices in graph such that is a subset of , is a subset of , and . In the following cases, we can be concluded that the metric dimension of is . Case 1: if , then we can assume that . Hence, there is a pair of distinct vertices , such that , and a shortest path from to is . Therefore, the metric representation of the vertices is the same as -vector, relative to . Thus, is not a resolving set of . Case 2: if and there are vertices , such that is adjacent to in , then there are vertices such that is adjacent to in . Therefore, the metric representation of the vertices is the same as -vector, relative to . Thus, is not a resolving set of . Case 3: now, let , and suppose that, for all the vertices in , we have is not adjacent to in , that is, . Also, for all the vertices in , we have is not adjacent to in , that is, . We may assume that and . So, we can assume that an arranged subset of vertices in graph is . Hence, , for .Therefore, the metric representations of the vertices relative to are the -vectors:and , ,...,. Thus, all the vertices in have different representations relative to . This implies that is a resolving set of .
Theorem 2. Ifis an even integer greater than or equalandis a Cayley graph on the dihedral group, whereis defined already, then the cardinality of minimum doubly resolving set ofis.
Proof. By the previous theorem, we know that the arranged subset of vertices in the graph is a resolving set for . We show that the subset is a doubly resolving set of . It is sufficient to show that, for two vertices and in graph , there are vertices such that . Consider two vertices and of . By the following cases, we can be concluded that the minimum cardinality of a doubly resolving set of is . Case 1: consider a pair of distinct vertices such that . Then, the length of a shortest path from to is or . Let be two vertices in such that a shortest path from to in graph is . We may assume that and . Hence, by taking and , we have . Therefore, the vertices and of doubly resolve . Now, let be two vertices in such that a shortest path from to in graph is . We may assume that and . Hence, by taking and , we have Therefore, the vertices and of doubly resolve . Case 2: consider a pair of distinct vertices such that and . Then, the length of a shortest path from to is or . Suppose a pair of distinct vertices and are adjacent in graph , that is, . We may assume that and . Hence, by taking and , we have . Therefore, the vertices and of doubly resolve . Now, suppose a pair of distinct vertices and are not adjacent in graph , that is, . We may assume that and . Hence, by taking and , we have . Therefore, the vertices and of doubly resolve . Case 3: consider a pair of distinct vertices such that and . Then, the length of a shortest path from to is or . We can show that the subset of vertices in graph is a doubly resolving set of . Because by Theorem 1, we can be concluded that is also a resolving set of
Lemma 1. Ifis an even integer greater than or equalandis a Cayley graph on the dihedral group, whereis defined already, then the subsetof vertices in graphis not a strong resolving set of.
Proof. We know that the arranged subset of vertices in graph is a resolving set for of size . Now, let , where , , and , where is a subset of and is a subset of . Consider two vertices in such that and is not adjacent to in , that is, . In the following cases, we show that there is not such that is strongly resolve vertices and . For every vertex , we have or . Case 1: if , then the length of a shortest path from to is or and length of a shortest path from to is or . Note that, if , then . Therefore, is not strongly resolve vertices and . In particular, if , then or , and hence, is not strongly resolve vertices and because . Case 2: if , then the length of a shortest path from to is and length of a shortest path from to is . Therefore, is not strongly resolve vertices and .
Theorem 3. Ifis an even integer greater than or equalandis a Cayley graph on the dihedral group, whereis defined already, then the strong metric dimension ofis.
Proof. Let , where and . It is not hard to see that if , then the size of largest clique in the graph is . Moreover, we know that the subset of vertices in is a clique in the graph . Now, let the subset of vertices in be . In the following cases, we show that the subset of vertices in is not a strong resolving set of . Case 1: let and . We know that and , and hence, for every such that , we have and . Thus, is not strongly resolve vertices and . Case 2: now, let and . We know that and , and hence, for every such that , we have and . Thus, is not strongly resolve vertices and .Therefore, the subset of the vertices in graph is not a strong resolving set of . From the above cases, we can be concluded that the minimum cardinality of a strong resolving set for must be .
4. Conclusion
Computing resolving parameters of a graph is an NP-hard problem. In this study, we considered a class of Toeplitz graphs and we denoted by so that they are isomorphic to the Cayley graph , which is defined already. In fact, this class of graphs is vertex transitive, and by calculating the spectrum of the adjacency matrix related with them, we showed that this class of graphs cannot be edge transitive. Also, we proved that this class of graphs cannot be distance regular, and because of the difficulty of the computing resolving parameters of a class of graphs which are not distance regular, we regarded this as justification for our focus on some resolving parameters. In particular, we determined the minimal resolving set, doubly resolving set, and strong metric dimension for this class of graphs.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
This work was supported in part by Natural Science Fund of Education Department of Anhui Province, under Grant KJ2020A0478.