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International Journal of Mathematics and Mathematical Sciences
Volume 2014 (2014), Article ID 308718, 4 pages
http://dx.doi.org/10.1155/2014/308718
Research Article

Product of Locally Primitive Graphs

Jundishapur University of Technology, Dezful 64616-18674, Iran

Received 21 January 2014; Revised 2 April 2014; Accepted 16 April 2014; Published 4 May 2014

Academic Editor: Seppo Hassi

Copyright © 2014 Amir Assari. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Many large graphs can be constructed from existing smaller graphs by using graph operations, such as the product of two graphs. Many properties of such large graphs are closely related to those of the corresponding smaller ones. In this paper we consider the product of two locally primitive graphs and prove that only tensor product of them will also be locally primitive.

1. Introduction

Let be a simple graph, where is the set of vertices and is the set of edges of . An edge joining the vertices and is denoted by . The group of automorphisms of is denoted by , which acts on vertices of . is called vertex transitive if acts transitively on the set of vertices. is called -locally transitive or -locally primitive for (or simply locally primitive or locally transitive when ) if acts transitively or primitively on , respectively, for each vertex , where is the set of vertices which are adjacent to and is the stabilizer of in . It is known that 2-arc-transitive graphs form a proper subclass of vertex transitive locally primitive graphs.

Let be a group and let be a nonempty subset of . The Cayley graph of with respect to is denoted by is defined as a graph with vertex set and is an edge of if and only if for some we have .

Many large graphs can be constructed by expanding of small graphs; thus it is important to know which properties of small graphs can be transferred to the expanded one; for example, Li et al. in [1] proved that the lexicographic product of vertex transitive graphs is also vertex transitive as well as the lexicographic product of edge transitive graphs, and Jaradat et al. in [2] found the basis number of the semicomposition product of two paths and a cycle with a path. Here we consider seven products of graphs as the expander graph which is described below, and hence when we talk about the product of graphs, we mean that the product is one of the following products.

Definition 1. Let and be two graphs. , the product of them, is a graph with vertex set , and the vertex is adjacent to in if one of the relevant conditions happens depending on the product. (1)Cartesian product: is adjacent to in and or and is adjacent to in .(2)Tensor product: is adjacent to in and is adjacent to in .(3)Strong product: is adjacent to in and or and is adjacent to in or is adjacent to in and is adjacent to in .(4)Lexicographic: is adjacent to in or and is adjacent to in .(5)Conormal product: is adjacent to in or is adjacent to in .(6)Modular product: is adjacent to in and is adjacent to in or is not adjacent to in and is not also adjacent to in .(7)Rooted product with root is adjacent to in and or and is adjacent to in .

Locally primitive Cayley graphs and 2-arc-transitive graphs have been extensively studied; see, for example, [38] and references therein. These motivated the author to investigate if the product of two locally transitive or locally primitive graphs has the property as well.

2. Main Results

Theorem 2. Let and be two simple graphs, let be a subgroup of , and let be a subgroup of . Then is also a subgroup of , where is the product of and .

Proof. Suppose . Thus is an automorphism of and is an automorphism of . is a bijection of and is a bijection of , which implies is also a bijection of , the vertex set of . Now assume and are two arbitrary adjoint vertices of . We distinguish seven kinds of products as follows.
Cartesian product: suppose is the Cartesian product of the graphs and . If and are adjoint in the Cartesian product, by the definition of Cartesian product, we have and is adjacent to in or vice versa.
For the case , and are adjoint in . is a map; thus . is an automorphism of ; hence is adjacent to in . Thus is adjacent to , which says is also adjacent to in .
Similar argument can be done for the case being adjacent to in and as well, which says preserves the edges of the graph .
Tensor product: suppose is tensor product of the graphs and . If and are adjoint in the graph , by the definition of tensor product, we imply that and are edges of and , respectively.
is an automorphism of the graph , implying is an edge of the graph . Being an automorphism of the graph yields will be an edge of ; that is, and are also adjacent in the graph ; that is, preserves the edges of the graph .
Strong product: from the case 1 and case 2 we can deduce that in this case we can also say preserves the edges of the graph .
Lexicographic product: if is the lexicographic product of the graphs and and and are joint by an edge in the graph , by the definition of lexicographic product, is an edge of , or and is adjacent to in .
If is adjacent to in , then is also adjacent to in , since is an automorphism of the graph , and hence by the definition of the lexicographic product, is adjacent to in the graph ; that is, and are joined by an edge in the graph .
For the case and being adjacent to in the same argument in case 1 can be done to show that is also adjacent to in the graph .
Conormal product: if is the conormal product of the graphs and and and are joint by an edge in the graph , by the definition of conormal product we have either or , where and are the edge set of the graphs and , respectively.
For the first case, since is an automorphism of , thus , and by the definition of conormal product, is adjacent to in the graph .
And for the case we will have , since is an automorphism of the graph . And hence is adjacent to in the graph .
Therefore in both cases we have that and are joint by an edge in the graph ; that is preserves the edges.
Modular product: we have for if and only if and , because and are automorphisms of the graphs and respectively. Therefore for if and only if and .
Now if is the modular product of the graphs and and and are joint by an edge in the graph , by the definition of modular product, we have , or for , where is the set of edges of .
For the first case we have and , implying is adjacent to in the graph .
For the latter case we have and ; that is is adjacent to in the graph .
And we conclude that preserve the edges.
Rooted product: similar argument of case can be done to prove that , the rooted product of graphs and , also preserve the edges, but we have considered that if is rooted product with root , then the image of it with respect to will be a rooted graph with root .
Thus if is one of the 7 kinds of products of and , then will be a bijection on which preserves the edges of the graph , implying is an automorphism of the graph .

Next lemma is simple to prove but useful in the literature. Thus we mention it without proof.

Lemma 3. For two simple graphs and and , If is the product of graphs and , then the neighbourhood of vertex of the vertex set of is as follows. (1)Cartesian product: .(2)Tensor product: .(3)Strong product: .(4)Lexicographic product: .(5)Conormal product: .(6)Modular product: , where is the complement set of in the vertex set for .(7)Rooted product with root : and , for .

Now we focus on simple graphs, by which we mean an undirected graph with no loops.

Theorem 4. Let and be - and -locally transitive nonempty (nonedgeless) simple graphs, respectively, then , the product of and , is also -locally transitive graph if and only if is the tensor product of them.

Proof. By the definition of locally transitive graph, we have to determine if is one of the seven kinds of graph product of and ; acts transitively on the set , where is any arbitrary element of the vertex set of and is the stabilizer of in .
If is in the stabilizer of in , then and we have Thus and , implying and ; that is, . The converse is also true which says
Let be Cartesian product of and . Lemma 3 implies that . Now for every we have , since is in the stabilizer of in . Similar argument shows that . and both are edgeless and hence the sets and are nonempty for some and which are disjoint from the sets and , respectively. Thus there is not any which sends to for some and , implying is not locally transitive.
Similar argument shows that if is strong product, lexicographic product, conormal product, or rooted product (consider for some in an edge of ); then it is not locally transitive graph.
For and we have , which implies can not be in ; that is, modular product of them can not be locally transitive.
If is the tensor product of and , then by Lemma 3, . Now if and are - and -locally transitive, respectively, then and act transitively on the set of and , respectively, for every and , and we conclude that acts transitively on the set for every ; that is, is locally transitive.

Lemma 5. Let and be two simple graphs which are - and -locally primitive, respectively; then is -locally primitive graph, where is the tensor product of and .

Proof. By Theorem 2, is a subgroup of the automorphism group of graph .
Suppose be an arbitrary vertex of the graph . By Lemma 3 we have .
is in stabilizer of in if and only if if and only if and ; that is, .
By assumption acts on primitively as well as on . Thus it is enough to show that acts primitively on . But transitivity of it arises from Theorem 4, and hence we should prove that it does not have any nontrivial block.
Suppose not, and take some nontrivial block of in . Set and , where is the projective map of th coordinate.
For , if , that is, exists such that for some we have , then for some , both of and belong to .
We also know acts on primitively and so transitively; thus we can say, for some , ; that is for some , and by permittivity condition should be the same as ; thus, for every , some exists such that implies ; therefore should belong to , and thus . But is bijection and so ; that is, is a block for .
Similarly, is also a block for . Thus if is a nontrivial block, then either or should also be a nontrivial block which is a contradiction to the assumption.

By the definition of locally permittivity and Lemma 5 and Theorem 4 we conclude the following theorem.

Theorem 6. Let and be two nonempty simple graphs which are - and -locally primitive, respectively, then , the product of and , is -locally primitive if and only if is the tensor product.

In [9], the authors proved the following lemma for semigroups and hence we can conclude it is valid for groups.

Lemma 7. Let and be two groups and let and be inverse closed subsets of them, respectively, which does not contain the identity element. If and , then the tensor product of the graphs and is also a Cayley graph of the group with respect to the subset .

Now by the Theorem 6 and Lemma 7 we can conclude the following theorem.

Theorem 8. Let and be two groups, let be an inverse closed subset of , and let be an inverse closed subset of and none of them have the identity. If is -locally primitive and is -locally primitive Cayley graph, then is locally primitive Cayley graph.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The author would like to thank the anonymous referee for his useful comments and suggestions.

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