Concerning Some Properties of Signed Graphs Associated With Specific Graphs
aa r X i v : . [ m a t h . C O ] S e p Concerning Some Properties of Signed GraphsAssociated With Specific Graphs
Y. Bagheri, A. R. Moghaddamfar and
F. Ramezani In memory of Professor Michael Neumann.
Abstract
Two signed graphs are called switching isomorphic if one of them is isomorphic to a switchingequivalent of the other. To determine the number of switching non-isomorphic signed graphs on aspecific graph, we will establish a method based on the action of its automorphism group. As anapplication and computational results, we classify all the switching non-isomorphic signed graphsarising from the complete graph K and the generalized Petersen graph GP(7 , All graphs considered in this paper are, simple and undirected. Let Γ be a graph of order n withvertex set V Γ = { v , v , . . . , v n } and edge set E Γ . For vertices v i , v j ∈ V Γ , we will write v i ∼ v j if v i v j ∈ E Γ . The adjacency matrix A = A (Γ) of a graph Γ of order n is the symmetric (0 , a ij ] n × n such that a ij = 1 if v i ∼ v j , and 0, otherwise.A signed graph (or sigraph ) is an ordered pair Σ = (Γ , σ ) consists of a simple graph Γ = ( V Γ , E Γ ),referred to as its underlying graph of Σ, and a sign function σ : E Γ → { +1 , − } . The concept of signedgraphs is given by Harary [5]. Recently there are some papers which have studied signed graphs fromdifferent perspectives (see for instance [2, 4, 8, 9, 10, 12, 13, 14]). We denote by S (Γ) the set of allsigned graphs with underlying graph Γ. For a signed graph Σ = (Γ , σ )), the vertex set V Σ of Σ coincidewith the vertex set of its underlying graph, while the edge set E Σ is divided into two disjoint subsets E +Σ and E − Σ (defined by σ ) that contain positive and negative edges, respectively. For a signed graphΣ = (Γ , σ ), by Σ + and Σ − , we mean the following unsigned graphs:Σ + = ( V Γ , E Σ + ) and Σ − = ( V Γ , E Σ − ) , where E Σ + = σ − (+1) and E Σ − = σ − ( − , Given a signed graph Σ = (Γ , σ ) with vertex set V Σ = V Γ = { v , v , . . . , v n } , the adjacency matrixof Σ is defined as a square matrix A σ = A (Σ) = [ a σij ] n × n with a σij = σ ( v i v j ) a ij , Corresponding author, email: [email protected]. a ij is the ( i, j )-entry of the adjacency matrix of the underlying graph Γ, or equivalently, a σij = , if v i ∼ v j and σ ( v i v j ) = +1, − , if v i ∼ v j and σ ( v i v j ) = − , if v i ≁ v j .A switching function is a function θ : V Γ → { +1 , − } . A signed graph Σ is transformed by a switchingfunction θ to a new signed graph Σ θ = (Γ , σ θ ) such that the underlying graph remains the same andthe sign function σ θ is defined on an edge v i v j ∈ E Γ by σ θ ( v i v j ) = θ ( v i ) σ ( v i v j ) θ ( v j ) . Two signed graphs Σ , Σ ∈ S (Γ) are said to be switching equivalent , denoted by Σ ∼ Σ , if thereexists a switching function θ such that Σ = Σ θ , or there exists a diagonal matrix D θ = diag( θ ( v ) , θ ( v ) , . . . , θ ( v n )) , such that A (Σ ) = D θ A (Σ ) D θ , otherwise they are called switching non-equivalent . Recall that switching of a signed graph at a vertex v reverses the sign of each edge incident with v . Hence, alternatively we might say that two signedgraphs Σ , Σ ∈ S (Γ) are switching equivalent if Σ is obtained from a series of switchings of Σ .It is clear that being ‘switching equivalent’ defines an equivalence relation on S (Γ), and thereforeit partitions the set S (Γ) into equivalence classes. We denote the equivalence class containing thesigned graph Σ ∈ S (Γ) by [Σ]. The set of all equivalence classes will be denoted by Ω s (Γ).Two signed graphs Σ = (Γ , σ ) and Σ = (Γ , σ ) are isomorphic , denoted by Σ ∼ = Σ , if thereis a graph isomorphism from Γ to Γ , which preserves the signs of edges. Similarly, the signed graphsΣ and Σ are said to be switching isomorphic if Σ is isomorphic to a signed graph which is switchingequivalent to Σ , otherwise we call them switching non-isomorphic .Given a graph Γ with n vertices and m edges, there are 2 m ways of constructing a signed graphon Γ. When Γ is connected, we show that there are 2 m − n +1 mutually switching non-equivalent signedgraphs in S (Γ). Zaslavsky in [14] proved that there are only six different signed Petersen graphs, upto switching isomorphism. Motivated by his results, we will investigate the same problem for otherlists of graphs. We also define an action of the automorphism group Aut(Γ) on the set Ω s (Γ). Usingsome well-known results from Group Theory, we count the number of switching non isomorphic signedgraphs. The main idea is determining orbit size of each class [Γ , σ ]. Classifying those signed graphswith a same underlying graph which are mutually switching non-isomorphic, significantly decreasethe number of signed graphs for studying their properties. For the notation and definitions on grouptheoretical aspects, we refer the reader to [6].We now introduce some further notation and definitions on Graph Theory. Two distinct edges e i and e j are said to be adjacent if and only if they have a common end vertex. Two edges of a graphare disjoint if they do not share a common vertex. Two graphs Γ = ( V Γ , E Γ ) and Γ = ( V Γ , E Γ )are isomorphic , denoted by Γ ∼ = Γ , if there exists a bijection ϕ : V Γ → V Γ such that, for everypair of vertices v i , v j ∈ V Γ , v i v j ∈ E Γ if and only if ϕ ( v i ) ϕ ( v j ) ∈ E Γ . An automorphism of a graphΓ is a graph isomorphism between Γ and itself. For a graph Γ, we denote by Aut(Γ) the set of allautomorphisms of Γ which forms a group under the operation of composition, and we call this groupthe automorphism group of Γ. 2he sequel of this paper is organized as follows: In Section 2 we provide some preparatory results.The main results are presented in Section 3. Finally, we find the number of signed graphs with theunderlying graph K and the generalized Petersen graph GP(7 , We start with the following classical result, which is proved in [8, Proposition 3.1]. We give here anelementary proof of this result to make the paper self-contained.
Lemma 2.1. ([8, Proposition 3.1])
Let Γ be a connected graph with n vertices and m edges. Then,there are m − n +1 mutually switching non-equivalent signed graphs with underlying graph Γ .Proof. We apply double counting to the set S (Γ) containing all signed graphs on Γ. On the one hand,there are 2 m possible sign functions on E Γ , and so |S (Γ) | = 2 m .On the other hand, let S ∗ (Γ) denote the set of mutually switching non-equivalent signed graphs.Recall that, there are 2 n possible switching functions on V Γ . Suppose that θ is a switching functionon V Γ and Σ = (Γ , σ ) ∈ S ∗ (Γ) is a signed graph. We first argue that Σ θ = Σ − θ . Indeed, we have σ − θ ( v i v j ) = ( − θ ( v i )) σ ( v i v j )( − θ ( v j )) = θ ( v i ) σ ( v i v j ) θ ( v j ) = σ θ ( v i v j ) , for every v i v j ∈ E Γ , as desired.Next, we claim that if θ = θ are two arbitrary switching functions on V Γ such that Σ θ = Σ θ ,then θ = − θ . Indeed, since θ = θ , there exists u ∈ V Γ with θ ( u ) = θ ( u ). Let v be an arbitraryvertex in V Γ \ { u } . Since Γ is connected, there exists a path between u and v in Γ, say u = w , w , w , . . . , w n − , w n = v. It now follows from the equality Σ θ = Σ θ that for every i = 1 , , . . . , n −
1, we have σ θ ( w i w i +1 ) = σ θ ( w i w i +1 ) , or equivalently, θ ( w i ) σ ( w i w i +1 ) θ ( w i +1 ) = θ ( w i ) σ ( w i w i +1 ) θ ( w i +1 ) , that is, θ ( w i ) θ ( w i +1 ) = θ ( w i ) θ ( w i +1 ) . Since θ ( w ) = θ ( w ), one obtains θ ( w ) = θ ( w ). Continuing with our argument for i =2 , , . . . , n −
1, we finally conclude that θ ( w n ) = θ ( w n ), that is θ ( v ) = θ ( v ). Since v was arbi-trary, it follows that θ = − θ , as claimed. Therefore, any signed graph Σ ∈ S ∗ (Γ) admits 2 n − switching non-equivalent signed graphs, which implies that |S (Γ) | = |S ∗ (Γ) | × n − . Finally, we conclude that 2 m = |S ∗ (Γ) | × n − , from which the lemma follows at once.In what follows we will let c denote the number of connected components of Γ. As an immediateconsequence of Lemma 2.1 we have: Corollary 2.2.
Let Γ be a graph with n vertices, m edges and c connected components. Then, thereare m − n + c mutually switching non-equivalent signed graphs with underlying graph Γ . roof. This is immediate from Lemma 2.1 and the fact that every graph is disjoint union of itsconnected components.We make use of the following lemma, due to T. Zaslavsky see [13], in our paper.
Lemma 2.3. ([13, T. Zaslavsky])
Two signed graphs Σ and Σ in S (Γ) are switching equivalent ifand only if the symmetric difference of E Σ − and E Σ − is an edge cut of Γ . The generalized Petersen graph
GP( n, k ), for n > k ⌊ ( n − / ⌋ , is a cubic graph onthe vertex set V GP( n,k ) = Z × Z n , for which the edge set is defined as follows: E GP( n,k ) = { (0 , j )(0 , j + 1) , (0 , j )(1 , j ) , (1 , j )(1 , j + k ) | j = 0 , , . . . , n − } , where all sums are taken modulo n . These graphs were introduced by Coxeter [1] and named byWatkins [11]. For instance, the generalized Petersen graph GP(7 ,
2) is depicted in Figure 1. We usuallycall vertices (0 , , (0 , , . . . , (0 , n − outer vertices and (1 , , (1 , , . . . , (1 , n − inner vertices . Notethat the outer and inner vertices are arranged on an outer circle and an inner circle, respectively. Fig. 1.
The generalized Petersen graph GP(7 , A ( n, k ) = Aut(GP( n, k )). It is clear that A ( n, k ) contains two automorphisms, α (the rotation ) and β (the reflection ), defined by α : ( i, j ) ( i, j + 1) and β : ( i, j ) ( i, − j ) , for ( i, j ) ∈ Z × Z n . Let γ be the permutation of vertices defined by γ : (0 , j ) (1 , kj ) and γ : (1 , j ) (0 , kj ). Again,it is not difficult to see that γ is an automorphism of GP( n, k ) exactly when k ≡ ± n ). Thefollowing theorem (which is taken from [3]), determines the automorphism group of a generalizedPetersen graph. Theorem 2.4. (see [3])
Let n and k be positive integers and ( n, k ) is not one of (4 , , (5 , , (8 , , (10 , , (10 , , (12 , or (24 , . Then, the following statements hold: (a) if k ≡ n ) , then we have A ( n, k ) = h α, β, γ | α n = β = γ = 1 , αβ = βα − , αγ = γα k , βγ = γβ i . (b) if k ≡ − n ) , then we have A ( n, k ) = h α, β, γ | α n = β = γ = 1 , αβ = βα − , αγ = γα k , βγ = γβ i . (c) if k
6≡ ± n ) , then we have A ( n, k ) = h α, β | α n = β = 1 , αβ = βα − i . .1 The action As is customary, we denote by Aut(Γ) the group of all automorphisms of a graph Γ. An automorphismof a signed graph Σ = (Γ , σ ) is an automorphism of Γ that preserves edge signs. The group of allautomorphisms of a signed graph Σ is denoted by Aut(Σ).We have the following easy observation (see Lemma 8.1 [14]):Aut(Σ) = Aut(Γ) ∩ Aut(Σ + ) = Aut(Γ) ∩ Aut(Σ − ) = Aut(Σ + ) ∩ Aut(Σ − ) . We now define an action of the automorphism group Aut(Γ) on the set Ω s (Γ) by setting[Γ , σ ] ϕ = [Γ , σ ϕ ] , where σ ϕ ( v i v j ) = σ ( ϕ − ( v i ) ϕ − ( v j )) . We need to verify that this action is well defined. For that we prove the image of any two switchingequivalent signed graphs, are also switching equivalent. If (Γ , σ ) and (Γ , σ ) are switching equivalent,then there exists a switching function θ such that (Γ , σ ) = (Γ , σ ) θ = (Γ , σ θ ), where as before σ θ ( v i v j ) = θ ( v i ) σ ( v i v j ) θ ( v j ) , for every v i v j ∈ E Γ . We now consider the switching function θϕ − : V Γ → { +1 , − } . We claim that(Γ , σ ϕ ) θϕ − = (Γ , σ ϕ ) , (1)which means that the signed graphs (Γ , σ ϕ ) and (Γ , σ ϕ ) are switching equivalent and we are done. Toprove (1), it suffices to show that (Γ , ( σ ϕ ) θϕ − ) = (Γ , σ ϕ ) , or equivalently, ( σ ϕ ) θϕ − = σ ϕ . In fact, for each v i v j ∈ E Γ , we have( σ ϕ ) θϕ − ( v i v j ) = θϕ − ( v i ) σ ϕ ( v i v j ) θϕ − ( v j )= θϕ − ( v i ) σ ( ϕ − ( v i ) ϕ − ( v j )) θϕ − ( v j )= σ θ ( ϕ − ( v i ) ϕ − ( v j ))= σ ( ϕ − ( v i ) ϕ − ( v j ))= σ ϕ ( v i v j ) , as required. Lemma 2.5.
Let Γ be a simple graph and let Aut(Γ) act on Ω s (Γ) as above. Two signed graphs (Γ , σ ) and (Γ , σ ) are switching isomorphic if and only if [Γ , σ ] and [Γ , σ ] belong to the same orbit.In particular, the number of switching non-isomorphic signed graphs is equal to the number of orbitsof this group action. roof. Let (Γ , σ ) and (Γ , σ ) be two switching isomorphic graphs. By the definition, we may assumethat (Γ , σ ) is isomorphic to (Γ , σ ) which is a switching equivalent to (Γ , σ ). Hence, there exists ϕ ∈ Aut(Γ), which preserves the sign of edges in (Γ , σ ), that is if v i v j ∈ E Γ , then σ ( v i v j ) = σ ( ϕ − ( v i ) ϕ − ( v j )) . This shows that σ = σ ϕ , and so[Γ , σ ] ϕ = [Γ , σ ϕ ] = [Γ , σ ] = [Γ , σ ] , which means that [Γ , σ ] and [Γ , σ ] are in the same orbit.Conversely, if [Γ , σ ] and [Γ , σ ] both lie in the same orbit of Ω s (Γ) under Aut(Γ), then there exists ϕ ∈ Aut(Γ) such that [Γ , σ ] = [Γ , σ ] ϕ , or equivalently, [Γ , σ ] = [Γ , σ ϕ ]. This means that (Γ , σ ) and(Γ , σ ϕ ) are switching equivalent, and since (Γ , σ ϕ ) ∼ = (Γ , σ ) ( ϕ is a corresponding isomorphism), itfollows that (Γ , σ ) and (Γ , σ ) are switching isomorphic. This completes the proof. Lemma 2.6.
Let
Σ = (Γ , σ ) be a signed graph and ϕ ∈ Aut(Γ) . Then Σ ϕ − and Σ − are isomorphic.Proof. We claim that ϕ : V Γ → V Γ is a graph isomorphism between Σ ϕ − and Σ − . Since ϕ is a bijectivemapping, it suffices to show that v i v j ∈ E Σ − if and only if ϕ ( v i ) ϕ ( v j ) ∈ E Σ ϕ − . Indeed, by definitionof σ ϕ , we have σ ϕ ( ϕ ( v i ) ϕ ( v j )) = σ ( v i v j ), and hence σ ( v i v j ) = − σ ϕ ( ϕ ( v i ) ϕ ( v j )) = − v i v j ∈ E Σ − if and only if ϕ ( v i ) ϕ ( v j ) ∈ E Σ ϕ − , as required. In the next result, we will determine a lower bound for the number of mutually switching non-isomorphic signed graphs on n vertices with a complete underlying graph. We need to introduce someterminology. Let Σ be a signed graph with vertex set V Σ = { v , v , . . . , v n } . We denote by d +Σ ( v i )(resp. d − Σ ( v i )) the number of positive (resp. negative) edges incident with v i , and denote by ψ ( n, ∆)the number of non-isomorphic graphs Γ on n vertices with ∆(Γ) ∆. Theorem 3.1.
The number of mutually switching non-isomorphic signed graphs with a completeunderlying graph on n > vertices is at least ψ ( n, ⌊ n ⌋ − .Proof. Let n > A = { Γ , Γ , . . . , Γ l } be the set of all mutually non-isomorphicgraphs on the vertex set { , , . . . , n } . Clearly, every Γ i is a subgraph of K n . It is obvious that twosigned graphs Σ and Σ in S ( K n ) are isomorphic if and only if the induced subgraphs Σ − and Σ − are isomorphic. For i = 1 , , . . . , l , we define a function σ i : K n → {− , +1 } by σ i ( e ) = ( − e ∈ E Γ i , +1 otherwise . We claim the signed graphs { ( K n , σ i ) | i = 1 , , . . . , l } are mutually switching non-isomorphic. Toprove the claim, we must show that for each i = j of { , , . . . , l } , the signed graphs ( K n , σ i ) and( K n , σ j ) cannot be switching isomorphic. First of all, note that the functions σ i have been chosenso that the signed graphs ( K n , σ i ) and ( K n , σ j ) cannot be isomorphic. We now prove that ( K n , σ j )cannot be isomorphic to any switching equivalent pair of ( K n , σ i ).6e make a few observations before going on to prove anything. Notice that any edge cut in K n induces a complete bipartite graph K n ,n where n = n + n . On the other hand, for each vertex v in the graph Γ j , j = 1 , , . . . , l , we have d − ( K n ,σ j ) ( v ) = deg Γ j ( v ) j n k − . (2)Now we return to the proof of the claim. By way of contradiction we assume that the signed graph( K n , σ j ) is isomorphic to a signed graph ( K n , σ ) which is switching equivalent to ( K n , σ i ). Since( K n , σ ) ∼ ( K n , σ i ), by the definition there exists a switching function θ such that ( K n , σ ) = ( K n , σ i ) θ .Let [ S, S ] be a edge cut in K n , which induces the complete bipartite graph K n ,n where n = | S | and n = | S | . We may assume without loss of generality that n ⌊ n ⌋ n , hence, in the graph K n ,n , the degree of each vertex in S is more than or equal to ⌊ n ⌋ . By Lemma 2.3 the symmetricdifference of the edge sets E ( K n ,σ ) − , E ( K n ,σ i ) − induce an edge cut which is considered to be K n ,n .Let Σ i = ( K n ,n , ˆ σ ), where ˆ σ = σ i | E ( K n ,n ) . Now, it follows for every vertex v in S that d +Σ i ( v ) = n − d − Σ i ( v ) > n − d − ( K n ,σ i ) ( v ) > n − j n k + 1 > j n k − j n k + 1 > j n k . Since switching of a signed graph at a vertex set S reverses the sign of each edge in [ S, S ], we obtain d − Σ θi ( v ) > j n k . On the other hand, singed graphs ( K n , σ j ) and ( K n , σ ) are isomorphic, which implies that d − ( K n ,σ j ) ( v ) = d − ( K n ,σ ) ( v ) = d − ( K n ,σ i ) θ ( v ) > d − Σ θi ( v ) > j n k , which contradicts (2) and completes the proof.In what follows, we focus our attention on the generalized Petersen graphs Λ = GP( p, k ) where p > k
6≡ ± p ). It follows by Theorem 2.4 thatAut(Λ) = A ( p, k ) = h α, β | α p = β = 1 , αβ = βα − i , which is a dihedral group of order 2 p . In group-theoretic terms, A ( p, k ) is a Frobenius group withkernel Z p and complement Z , hence A ( p, k ) ∼ = Z p ⋊ Z . Here we shall be interested only in derivingelementary properties of automorphism groups of signed graphs on GP( p, k ). To do this, we introducea little more notation.If Ψ = (Λ , σ ) is a signed graph on Λ, then for each ∅ 6 = F ⊆ E Λ , define σ ( F ) := { σ ( e ) | e ∈ F } ⊆ { +1 , − } . Put E := { (0 , j )(0 , j + 1) | j = 0 , , . . . , p − } ,E := { (1 , j )(1 , j + k ) | j = 0 , , . . . , p − } ,E := { (0 , j )(1 , j ) | j = 0 , , . . . , p − } . We make a preliminary observation: h α i acts transitively on E i for each i , 1 i
3. For instance,suppose that e i = (0 , i )(1 , i + k ) and e j = (0 , j )(1 , j + k ) are two arbitrary elements of E such that j > i . We now consider the rotation α j − i . It follows by the definition of α that α j − i (0 , i ) = (0 , j − i + i ) = (0 , j ) and α j − i (1 , i + k ) = (1 , j − i + i + k ) = (1 , j + k ) , and so α j − i ( e i ) = e j . With this preliminary observation, we can now prove the following result.7 heorem 3.2. With the above notation, if | σ ( E ) | = | σ ( E ) | = | σ ( E ) | = 1 , then Aut(Ψ) = A ( p, k ) ,otherwise Aut(Ψ) contains at most two elements.Proof.
We already know that Aut(Ψ) = A ( p, k ) ∩ Aut(Ψ − ). Observe, first of all, that A ( p, k ) = { , α, α , . . . , α p − , β, αβ, α β, . . . , α p − β } , all rotations α, α , . . . , α p − have order p and all reflections β, αβ, . . . , α p − β have order 2. Obviously,Aut(Ψ) as a subgroup of A ( p, k ) has order 1, 2, p or 2 p . Since h α i acts transitively on E i for each i , none of these rotations is contained in Aut(Ψ − ), except when | σ ( E ) | = | σ ( E ) | = | σ ( E ) | = 1.As a matter of fact, in the case when | σ ( E ) | = | σ ( E ) | = | σ ( E ) | = 1, since the sets E , E and E are h α i -invariant, h α i ⊆ Aut(Ψ − ). Similarly, each of these sets is invariant under elements of theform α i β , i = 0 , , . . . , p −
1. Again it follows that if | σ ( E ) | = | σ ( E ) | = | σ ( E ) | = 1, then Aut(Ψ − )contains all elements of the form α i β , i = 0 , , . . . , p −
1. In this situation, we therefore conclude thatAut(Ψ − ) = A ( p, k ) = Aut(Ψ), as required.In what follows, we will focus our attention on signed graphs in which the underlying graph isa complete graph or a generalized Petersen graph. Let us first consider the signed graphs with theunderlying graph K n . In fact, the number of signed graphs with on K n is equal to |S ( K n ) | = 2( n ),and among all such signed graphs, by Lemma 2.1, the number of mutually switching non-equivalentsigned graphs with the underlying graph K n ie equal to | Ω s ( K n ) | = 2( n ) − n +1 = 2 ( n − n − / . The problem of finding the number of switching non-isomorphic graphs on K n is also useful as wellas interesting. Recall that if we consider the action of Aut( K n ) on Ω s ( K n ) by [ K n , σ ] ϕ = [ K n , σ ϕ ],where σ ϕ ( v i v j ) = σ ( ϕ − ( v i ) ϕ − ( v j )), then by Lemma 2.5, the number of switching non-isomorphicgraphs on K n is equal to the number of orbits of this action. It is well known that the automorphismgroup of the complete graph on n vertices Aut( K n ) is isomorphic to S n , which is n -fold transitive onthe set { , , . . . , n } for any n . Clearly n -fold transitivity implies k -fold transitive for 1 k n , soany triply transitivity implies doubly transitive and transitive. Thus, when n >
3, Aut( K n ) ∼ = S n isdoubly transitive, which means that Aut( K n ) is edge-transitive. Lemma 3.3.
Let
Aut( K n ) act on Ω s ( K n ) as above, and [ K n , σ ] ∈ Ω s ( K n ) . Then, we have [ K n , σ ] Aut( K n ) = (cid:8) [ K n , σ ′ ] | ( K n , σ ′ ) − ∼ = ( K n , σ ) − (cid:9) . Proof.
We put O := [ K n , σ ] Aut( K n ) and A := { [ K n , σ ′ ] | ( K n , σ ′ ) − ∼ = ( K n , σ ) − } . Assume first that[ K n , σ ′ ] ∈ A and ϕ : V Γ → V Γ is a graph isomorphism between ( K n , σ ) − and ( K n , σ ′ ) − . Obviously ϕ ∈ Aut( K n ). We claim that [ K n , σ ′ ] = [ K n , σ ] ϕ ∈ O . It suffices to show that ( K n , σ ′ ) = ( K n , σ ) ϕ ,or equivalently, ( K n , σ ′ ) = ( K n , σ ϕ ). Note that the last equality is also equivalent to σ ′ = σ ϕ . Since ϕ : V Γ → V Γ is a graph isomorphism between ( K n , σ ) − and ( K n , σ ′ ) − , it follows that v i v j ∈ E ( K n ,σ ) − if and only if ϕ ( v i ) ϕ ( v j ) ∈ E ( K n ,σ ′ ) − , or equivalently, σ ( v i v j ) = − σ ′ ( ϕ ( v i ) ϕ ( v j )) = − ϕ − ( v i ) for v i and ϕ − ( v j ) for v j in the last expression, we obtain σ ( ϕ − ( v i ) ϕ − ( v j )) = − σ ′ ( v i v j ) = −
1, or equivalently, σ ϕ ( v i v j ) = − σ ′ ( v i v j ) = −
1. This meansthat σ ϕ = σ ′ , as required.Assume next that [ K n , σ ] ϕ ∈ O . To prove [ K n , σ ] ϕ ∈ A , we must show that ( K n , σ ) ϕ − ∼ = ( K n , σ ) − ,or equivalently, ( K n , σ ϕ ) − ∼ = ( K n , σ ) − . We now consider the automorphism ϕ : V Γ → V Γ and notethat if v i v j ∈ E ( K n ,σ ) − , then σ ( v i v j ) = −
1. But then it is immediate that σ ϕ ( ϕ ( v i ) ϕ ( v j )) = σ (cid:0) ϕ − ( ϕ ( v i )) ϕ − ( ϕ ( v j )) (cid:1) = σ ( v i v j ) = − , ϕ ( v i ) ϕ ( v j ) ∈ E ( K n ,σ ϕ ) − . Obviously, the reverse direction also holds true. Therefore, we concludethat ϕ is an isomorphism between ( K n , σ ϕ ) − and ( K n , σ ) − , as required. In the following result, we focus our attention on a special case, that is K . We will use the followingnotation regarding the signed graphs: µ [Γ , σ ] = min Σ ∈ [Γ ,σ ] | E Σ − | . In this section, the negative and positive edges are drawn in red and blue lines, respectively.
Theorem 4.1. (see [7])
There are exactly seven signed graphs with the underlying graph K up toswitching isomorphism.Proof. First of all, we observe that by the preceding paragraph |S ( K ) | = 2 and | Ω s ( K ) | = 2 . LetAut( K ) act on Ω s ( K ) by [ K , σ ] ϕ = [ K , σ ϕ ], where σ ϕ ( v i v j ) = σ ( ϕ − ( v i ) ϕ − ( v j )). We shall tryto determine the number of orbits of this action instead of the number of switching non-isomorphicgraphs on K . More precisely, we shall show that 2 classes in Ω s ( K ) are partitioned into the 7 orbits O , O , . . . , O , of size 1, 10, 15, 12, 15, 10 and 1, respectively, corresponding to the signed graphsgiven in Figure 2. ( ) . ) . ) . ) . ) . ) . ) . ) . ) . ) . ) . ) . ) . ) . ( ) . ) . ) . ) . ) . ) . ) . ) . ) . ) . ) . ) . ) . ) . ( ) . ) . ) . ) . ) . ) . ) . ) . ) . ) . ) . ) . ) . ) . ( ) . ) . ) . ) . ) . ) . ) . ) . ) . ) . ) . ) . ) . ) . ( ) . ) . ) . ) . ) . ) . ) . ) . ) . ) . ) . ) . ) . ) . ( ) . ) . ) . ) . ) . ) . ) . ) . ) . ) . ) . ) . ) . ) . ( ) . ) . ) . ) . ) . ) . ) . ) . ) . ) . ) . ) . ) . ) . Fig. 2.
Seven switching non-isomorphic signed graphs on K .Before we continue with the proof of this result we make some general remarks. For signed graph( K , σ i ), it follows that O i = [ K , σ i ] Aut( K ) = { [ K , σ i ] ϕ | ϕ ∈ Aut( K ) } = { [ K , σ ϕi ] | ϕ ∈ Aut( K ) } (by the definition)= { [ K , σ ] | ( K , σ ) − ∼ = ( K , σ i ) − } . (by Lemma 3.3) (3)9ote that Aut( K ) is isomorphic to S which acts on the vertex-set 5-transitive, and so K is edge-transitive, that is, if v i v j , v k v l ∈ E K , then there exists an automorphism ϕ in Aut( K ) such that ϕ ( v i v j ) = v k v l . We can now return and finish off our proof. We treat separately the different cases:(a) µ [ K , σ ] = 0. The signed graph ( K , σ ) is depicted in Fig. 2 (1). In this case, ( K , σ ) − is thenull graph, that is, a graph with no edges. We see from (3) that O contains exactly one class[ K , σ ].(b) µ [ K , σ ] = 1. The signed graph ( K , σ ) is shown in Fig. 2 (2). In this case, ( K , σ ) − consistsexactly of three isolated vertices and hence O contains exactly 10 (one for each edge) class[ K , σ ] for which | σ − ( − | = 1.(c) µ [ K , σ ] = 2. We distinguish between the cases: ( K , σ ) − contains two adjacent edges or twodisjoint edges.(c.1) ( K , σ ) − contains two adjacent edges. In this case, the signed graph appears as shown inFig. 2 (3). Therefore, in veiw of (3), we need to count the number of signed graphs ( K , σ )such that ( K , σ ) − ∼ = ( K , σ ) − . It is now easy to check that the number of such signedgraphs is 12 (cid:18)(cid:18) (cid:19) × (cid:18) (cid:19)(cid:19) = 15 . Note that by switching the vertex of degree 2 in ( K , σ ) − we obtain a switching equivalentsigned graph of the same type.(c.2) ( K , σ ) − contains two disjoint edges. The corresponding signed graph in this case is asshown in Fig. 2 (5). As before, we conclude similarly that there are (cid:0)(cid:0) (cid:1) × (cid:0) (cid:1)(cid:1) = 15signed graphs ( K , σ ) such that ( K , σ ) − ∼ = ( K , σ ) − .(d) µ [ K , σ ] = 3. Here, we distinguish among three cases: the number of vertices of degree 2 is 3,2 or 1.(d.1) ( K , σ ) − contains three vertices of degree . The signed graph ( K , σ ) in this case isshown in Fig. 2(6). A routine argument shows that the number of signed graphs ( K , σ )such that ( K , σ ) − ∼ = ( K , σ ) − is (cid:0) (cid:1) = 10.(d.2) ( K , σ ) − contains two vertices of degree . In this case, ( K , σ ) − is a path of length 3 asshown in Fig. 2 (4). On the one hand, the number of such paths is (5 × × × / v v v v is a path of length 3, then by switching on { v } , { v } , { v , v } or { v , v } we obtain a switching equivalent signed graph of the same type. Hence,in this case, we have 60 / K , σ ) such that( K , σ ) − ∼ = ( K , σ ) − .(d.3) ( K , σ ) − contains one vertices of degree . In this case, by switching the vertex of degree2, we obtain a switching equivalent signed graph which has considered in (d.1).(e) µ [ K , σ ] = 4. The signed graph ( K , σ ) is depicted in Fig. 2 (7). If ( K , σ ) − ∼ = ( K , σ ) − , thenit follows by switching at vertices of degree 1 that both of ( K , σ ) and ( K , σ ) are switchingequivalent with ( K , τ ) for which τ − ( −
1) = E K . Hence, we conclude that [ K , σ ] = [ K , σ ],which shows that O contains exactly one class [ K , σ ].10t follows from this discussion that the set Ω s ( K ) can be partitioned into the 7 orbits O , . . . , O ,as required.Similarly, we have the following theorem. Theorem 4.2.
There are exactly signed graphs with the underlying graph GP(7 , up to switchingisomorphism.Proof. The proof is quite similar to the proof of Theorem 4.1, so we avoid here full explanation of alldetails. Let Γ = GP(7 , |S (Γ) | = 2 and | Ω s (Γ) | = 2 . Let Aut(Γ) acton Ω s (Γ) by [Γ , σ ] ϕ = [Γ , σ ϕ ]. Again, we determine the number of orbits of this action instead of thenumber of switching non-isomorphic graphs on Γ. As a matter of fact, the 2 classes in Ω s (Γ) can bepartitioned into the 36 orbits O , O , . . . , O , of size 1, 7, 7, 7, 7, 7, 7, 7, 7, 14, 7, 7, 7, 7, 14, 7, 14,7, 7, 7, 7, 1, 7, 7, 7, 7, 7, 7, 1, 7, 7, 7, 7, 14, 1 and 7, respectively, corresponding to the signed graphsgiven in Figure 3. Fig. 3.
36 switching non-isomorphic signed graphs on GP(7 , ( ) . ( ) . ( ) . ( ) . ( ) . ( ) . ( ) . ( ) . ( ) . ( ) . ( ) . ( ) . ( ) . ( ) . ( ) . ( ) . ) . ( ) . ( ) . ( ) . ( ) . ( ) . ( ) . ( ) . ( ) . ( ) . ( ) . ( ) . ( ) . ( ) . ( ) . ( ) . ( ) . ( ) . ( ) . ( ) . References [1] H. S. M. Coxeter, Self-dual configurations and regular graphs,
Bull. Amer. Math. Soc , 56 (5)(1950), 413–455.[2] B. Et-Taoui and A. Fruchard, On switching classes of graphs, (English summary)
Linear AlgebraAppl. , 549 (2018), 246–255.[3] R. Frucht, J. E. Graver, and M. E. Watkins, The groups of the generalized Petersen graphs,
Proc. Cambridge Philos. Soc. , 70(1971), 211–218.124] K. A. Germina, K. Hameed and S. T. Zaslavsky, On products and line graphs of signed graphs,their eigenvalues and energy, (English summary)
Linear Algebra Appl. , 435(10) (2011), 2432–2450.[5] F. Harary, On the notion of balance of a signed graph,
Michigan Math. J. , 2 (1953), 143–46.[6] I. M. Isaacs,
Finite Group Theory , Graduate Studies in Mathematics, 92. American Mathemat-ical Society, Providence, RI, 2008.[7] C. L. Mallows and N. J. A. Sloane, Two-graphs, switching classes and Euler graphs are equalin number,
SIAM J. Appl. Math. , 28 (1975), 876–880.[8] R. Naserasr, E. Rollov´a and ´E. Sopena, Homomorphisms of signed graphs,
J. Graph Theory ,79(3) (2015), 178–212.[9] H. Qi, T. L. Wong and X. Zhu, Chromatic number and orientations of graphs and signedGraphs,
Taiwanese J. Math. , 23(4) (2019), 767–776.[10] T. Soza´nski, Enumeration of weak isomorphism classes of signed graphs,
J. Graph Theory , 4(2)(1980), 127–144.[11] M. E. Watkins. A theorem on Tait colorings with an application to the generalized Petersengraphs. J. Combin. Theory 6 (2) (1969), 152–164.[12] G. Yu, L. Feng, and H. Qu, Signed graphs with small positive index of inertia, (English sum-mary)
Electron. J. Linear Algebra , 31 (2016), 232–243.[13] T. Zaslavsky, Signed graphs,
Discrete Appl. Math. , 4(1)(1982), 47–74.[14] T. Zaslavsky, Six signed Petersen graphs, and their automorphisms,
Discrete Math.
Yousef Bagheri, Ali Reza Moghaddamfar and Farzaneh RamezaniFaculty of Mathematics, K. N. Toosi University of Technology, P. O. Box – ,Tehran, Iran, E-mail addresses : [email protected] , [email protected]@kntu.ac.ir