JMATH Journal of Mathematics 2314-4785 2314-4629 Hindawi 10.1155/2017/6454736 6454736 Research Article An Interesting Property of a Class of Circulant Graphs Mirafzal Seyed Morteza 1 http://orcid.org/0000-0001-9758-8821 Zafari Ali 1 Bauer Michel Department of Mathematics Lorestan University Khoramabad Iran lu.ac.ir 2017 2722017 2017 22 07 2016 22 01 2017 2722017 2017 Copyright © 2017 Seyed Morteza Mirafzal and Ali Zafari. 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.

Suppose that Π=Cay(Zn,Ω) and Λ=Cay(Zn,Ψm) are two Cayley graphs on the cyclic additive group Zn, where n is an even integer, m=n/2+1, Ω=tZnt  is  odd, and Ψm=Ω{n/2} are the inverse-closed subsets of Zn-0. In this paper, it is shown that Π is a distance-transitive graph, and, by this fact, we determine the adjacency matrix spectrum of Π. Finally, we show that if n8 and n/2 is an even integer, then the adjacency matrix spectrum of Λ is n/2+11, 1-n/21, 1n-4/2, -1n/2 (we write multiplicities as exponents).

1. Introduction

In this paper, graph Γ=(V,E) always means a simple connected graph with n vertices (without loops, multiple edges, and isolated vertices), where V=V(Γ) is the vertex set and E=E(Γ) is the edge set. Graph Γ is called a vertex-transitive graph, if, for any x,yV, there is some π in Aut(Γ), the automorphism group of Γ, such that π(x)=y. Let Γ be a graph, the complement Γ¯ of Γ is the graph whose vertex set is V(Γ) and whose edges are the pairs of nonadjacent vertices of Γ. It is well known that, for any graph Γ, Aut(Γ)=Aut(Γ¯) . If Γ is a connected graph and (u,v) denotes the distance in Γ between the vertices u and v, then, for any automorphism π in Aut(Γ), we have (u,v)=(π(u),π(v)).

Let Υ=γ1,,γk+1 be a set and K be a group; then, writing Fun(Υ,K) to denote the set of all functions from Υ into K, we can turn Fun(Υ,K) into a group by defining a product: (1)fgγ=fγgγf,gFunΥ,K  and  γΥ, where the product on the right is in K. Since Υ is finite then the group Fun(Υ,K) is isomorphic to Kk+1 (a direct product of k+1 copies of K) via the isomorphism f(f(γ1),,f(γk+1)). Let H and K be groups and suppose that H acts on the nonempty set Υ. Then, the wreath product of K by H with respect to this action is defined to be the semidirect product Fun(Υ,K)H where H acts on the group Fun(Υ,K) via (2)fxγ=fγx-1fFunΥ,K,γΥ  and  xH. We denote this group by KwrΥH. Consider the wreath product G=KwrΥH. If K acts on a set Δ, then we can define an action of G on Δ×Υ by (3)δ,γf,h=δfγ,γhδ,γΔ×Υ, where (f,h)Fun(Υ,K)H=KwrΥH .

Let G be a group and HG a subgroup of G and SG. The Schreier coset graph on G/H generated by S is the graph Γ=Γ(G,H,S) with V(Γ)=G/H=gHgG the set of left cosets of H, and there is an edge (gH,sgH) for each coset gH and each sS. If S is inverse-closed, then Γ is an undirected multigraph (possibly with loops). Note that if 1G is the identity element of G, then Γ(G,1G,S)=Γ(G,S) is the Cayley graph on G generated by S. It is well known that every Cayley graph is vertex-transitive .

Let Γ be a graph with automorphism group Aut(Γ). Say that Γ is symmetric graph if, for all vertices u,v,x,y of Γ such that u and v are adjacent, also, x and y are adjacent, and there is an automorphism π in Aut(Γ) such that π(u)=x and π(v)=y. We say that Γ is distance-transitive if, for all vertices u,v,x,y of Γ such that (u,v)=(x,y), there is an automorphism π in Aut(Γ) satisfying π(u)=x and π(v)=y . It is clear that hierarchy of the conditions is(4)distance-transitivesymmetricvertex-transitive.

Eigenvalues of an undirected graph Γ are the eigenvalues of an arbitrary adjacency matrix of Γ. Harary and Schwenk  defined Γ to be integral, if all of its eigenvalues are integers. For a survey of integral graphs, see . In , the number of integral graphs on n vertices is estimated. Known characterizations of integral graphs are restricted to certain graph classes; see .

In this paper, suppose Π=Cay(Zn,Ω) and Λ=Cay(Zn,Ψm) are two Cayley graphs on the cyclic additive group Zn, where n is an even integer, m=n/2+1, Ω=tZnt  is  odd, and Ψm=Ω{n/2} are the inverse-closed subsets of Zn-{0}. One of our goals in this paper is to obtain all eigenvalues of the Cayley graph Λ=Cay(Zn,Ψm). First, we determine the group automorphism of Π and we show that Π is a distance transitive graph; also, by this fact, we determine the adjacency matrix spectrum of Π. Finally, according to these facts, we show that if n8 and n/2 is an even integer, then the adjacency matrix spectrum of Λ is n/2+11,1-n/21,1n-4/2,-1n/2 (we write multiplicities as exponents).

2. Definitions and Preliminaries Definition 1 (see [<xref ref-type="bibr" rid="B5">3</xref>, <xref ref-type="bibr" rid="B6">8</xref>]).

For any vertex v of a connected graph Γ, one defines (5)Γrv=uVΓu,v=r,where r is a nonnegative integer not exceeding d, the diameter of Γ. It is clear that Γ0(v)={v}, and V(Γ) is partitioned into the disjoint subsets Γ0(v),,Γd(v), for each v in V(Γ). The graph Γ is called distance-regular with diameter d and intersection array {b0,,bd-1;c1,,cd}, if it is regular of valency k and, for any two vertices u and v in Γ at distance r, one has Γr+1vΓ1u=br, and Γr-1vΓ1u=cr, 0rd. The numbers cr,br, and ar, where (6)ar=k-br-cr0rd, is the number of neighbours of u in Γr(v) for (u,v)=r, are called the intersection numbers of Γ. Clearly b0=k, bd=c0=0, and c1=1.

Remark 2 (see [<xref ref-type="bibr" rid="B5">3</xref>]).

It is clear that if Γ is distance-transitive graph, then Γ is distance-regular.

Lemma 3 (see [<xref ref-type="bibr" rid="B5">3</xref>]).

A connected graph Γ with diameter d and automorphism group G=Aut(Γ) is distance-transitive if and only if it is vertex-transitive and the vertex-stabilizer Gv is transitive on the set Γr(v), for each r0,1,,d, and vV(Γ).

Theorem 4 (see [<xref ref-type="bibr" rid="B6">8</xref>]).

Let Γ be a distance-regular graph which the valency of each vertex as k, with diameter d, adjacency matrix A, and intersection array, is(7)b0,b1,,bd-1;c1,c2,,cd. Then, the tridiagonal (d+1)×(d+1) matrix(8)jΓ=a0b000c1a1b100c2a2b2cd-2ad-2bd-200cd-1ad-1bd-100cdad determines all the eigenvalues of Γ.

Theorem 5 (see [<xref ref-type="bibr" rid="B7">9</xref>]).

Let F be a field and let R be a commutative subring of Fn×n, the set of all n×n matrices over F. Let MRm×m, then detF(M)=detF(detR(M)).

Theorem 6 (see [<xref ref-type="bibr" rid="B4">10</xref>]).

Let Γ be a graph such that contains k+1 components Γ1,,Γk+1. If, for any iI={1,,k+1}, ΓiΓ1, then Aut(Γ)Aut(Γ1)wrISym(k+1).

2.1. Main Results Proposition 7.

Let Π=Cay(Zn,Ω) be the Cayley graph on the cyclic group Zn(n4), where Ω=tZnt  is  odd is the inverse-closed subset of Zn-{0}. Then Aut(Π)Sym(n/2)wrISym(2), where I={1,2}.

Proof.

Let V(Π)=1,,n be the vertex set of Π. By assumption, the size of the every independent set of vertices in Π is n/2, because Π is a vertex-transitive graph and the size of every clique in graph Π is 2. Therefore, for any xV(Π), there is exactly n/2, yV(Π) such that x-1yΩ. Hence, if x-1yΩ, then two vertices x and y are adjacent in the complement Π¯ of Π, so Π¯ contains 2 components Π1,Π2 such that Π1Π2Kn/2, where Kn/2 is the complete graph of n/2 vertices. Therefore, Π¯2Kn/2. Hence, by Theorem 6, Aut(Π¯)Aut(Kn/2)wrISym(2)=Sym(n/2)wrISym(2).

Proposition 8.

Let Π=Cay(Zn,Ω) be the Cayley graph on the cyclic group Zn(n4), where n is an even integer and Ω=tZnt  is  odd is the inverse-closed subset of Zn-{0}; then Π is a distance-transitive graph.

Proof.

Suppose that u,v,x,y are vertices of Π such that (u,v)=(x,y)=r, where r is a nonnegative integer not exceeding d, the diameter of Π. So (u,v)=(x,y)=1 or 2, since d=2.

(a) If (u,v)=(x,y)=2, then u-1vΩ and x-1yΩ. Therefore, two vertices u and v are adjacent in the complement Π¯ of Π, also two vertices x and y are adjacent in the complement Π¯ of Π. So Π¯ contains 2 components Π1,Π2 such that Π1Π2Kn/2. Therefore Π¯2Kn/2; hence we may assume π=(ux)(vy)Aut(Π¯)=Aut(Π), so π(u)=x and π(v)=y.

(b) If (u,v)=(x,y)=1, then, by Lemma 3, it is sufficient to show that vertex-stabilizer Gv is transitive on set Πr(v) for every r0,1,2 and every vV(Π), because Π is a vertex-transitive graph. In this case, let V(Π)=1,2,,n be the vertex set of Π and G=Aut(Π). Consider the vertex v=1 in V(Π), then Π0(v)={1}, Π1(v)=tZnt  is  even, and Π2(v)=1tZnt  is  odd. Let H be the group that is generated by all elements of sets Π1(v) and Π2(v), say H=(2,4,,n),3,5,,n-1. It is clear that H is a subgroup of Aut(Π), so the group H is a subgroup of Gv such that transitive on the set Πr(v) for each r0,1,2. Note that if 1vV(Π), then, we can show that vertex-stabilizer Gv is transitive on the set Πr(v) for each r0,1,2, because Π is a vertex-transitive graph.

Proposition 9.

Let Π=Cay(Zn,Ω) be the Cayley graph on the cyclic group Zn(n4), where n is an even integer and Ω=tZnt  is  odd is the inverse-closed subset of Zn-{0}; then Π is an integral graph.

Proof.

By Remark 2, it is clear that Π is distance-regular, because Π is a distance-transitive graph. Let V(Π)={1,2,,n} be the vertex set of Π. Consider the vertex v=1 in V(Π); then Π0(v)={1}, Π1(v)=tZnt  is  even, and Π2(v)=1tZnt  is  odd. Let be u in V(Π) such that (u,v)=0; then u=v=1 and Π1vΠ1u=n/2; hence b0=n/2 and, by Definition 1, a0=n/2-b0=0. Also, if u in V(Π) and (u,v)=1, then two vertices u,v are adjacent in Π, so Π0vΠ1u=1 and Π2vΠ1u=n/2-1; hence c1=1, b1=n/2-1, and a1=n/2-b1-c1=0. Finally, if u in V(Π) and (u,v)=2, then two vertices u,v are not adjacent in Π, so Π1vΠ1u=n/2; hence c2=n/2 and a2=n/2-c2=0. So the intersection array of Π is {n/2,n/2-1;1,n/2}. Therefore, by Theorem 4, the tridiagonal (3)×(3) matrix(9)a0b00c1a1b10c2a2=0n2010n2-10n20determines all the eigenvalues of Π. It is clear that all the eigenvalues of Π are n/2,-n/2,0, and their multiplicities are 1,1,n-2, respectively. So Π is an integral graph.

Conclusion 10.

Let Π=Cay(Zn,Ω) be the Cayley graph on the cyclic group Zn as before with the adjacency matrix M=AAAA, and characteristic polynomial ΦΠ(x) then is(10)Φ2Ax=detxIn/2-2A=x-n2x+n2xn-4/2.

Proof.

It is easy to show that the adjacency matrix M=AAAA, where A is n/2×n/2 matrix; hence, by Proposition 9 and Theorem 5, (11)ΦΠx=detxIn-M=detxIn/2-A-A-AxIn/2-A=detxIn/2-A2-A2=detxIn/2-A-AdetxIn/2-A+A=detxIn/2-2AdetxIn/2=Φ2Axxn/2=x-n2x+n2xn-4/2xn/2.

Proposition 11.

Let Λ=Cay(Zn,Ψm) be the Cayley graph on the cyclic group Zn as before with the adjacency matrix M^ and characteristic polynomial ΦΛ(x). If n8 and n/2 is an even integer, then (12)ΦΛx=x-n2+1x-1-n2x-1n-4/2x+1n/2.

Proof.

It is easy to show that the adjacency matrix M^=AA+IA+IA, where A is n/2×n/2 matrix; hence, by Conclusion 10 and Theorem 5,(13)ΦΛx=detxIn-M^=detxIn/2-A-A+I-A+IxIn/2-A=detxIn/2-A2-A+I2=detxIn/2-A-A+IdetxIn/2-A+A+I=detx-1In/2-2Adetx+1In/2=Φ2Ax-1x+1n/2=x-n2+1x-1-n2x-1n-4/2x+1n/2.

Conclusion 12.

Let Λ=Cay(Zn,Ψm) be the Cayley graph on the cyclic group Zn, where n is an even integer, m=n/2+1, and Ψm=Ω{n/2} is the inverse-closed subset of Zn-{0}. If n8 and n/2 is an even integer, then the adjacency matrix spectrum of Λ is n/2+11, 1-n/21, 1n-4/2, -1n/2.

Competing Interests

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

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