McKay trees

Given a finite group $G$ and its representation $\rho$, the corresponding McKay graph is a graph $\Gamma(G,\rho)$ whose vertices are the irreducible representations of $G$; the number of edges between two vertices $\pi,\tau$ of $\Gamma(G,\rho)$ is $dim Hom_G(\pi \otimes \rho, \tau) $. The collection of all McKay graphs for a given group $G$ encodes, in a sense, its character table. Such graphs were also used by McKay to provide a bijection between the finite subgroups of $SU(2)$ and the affine Dynkin diagrams of types $A, D, E$, the bijection given by considering the appropriate McKay graphs. In this paper, we classify all (undirected) trees which are McKay graphs of finite groups and describe the corresponding pairs $(G,\rho)$; this classification turns out to be very concise. Moreover, we give a partial classification of McKay graphs which are forests, and construct some non-trivial examples of such forests.


Introduction
1.1.Definition of the McKay graph.For a finite group G and its complex representation ρ, we consider the McKay graph Γ(G, ρ): its vertices are given by the set Irr(G) of irreducible complex representations of G, and the number of edges N π,τ between two vertices π, τ ∈ Irr(G) of Γ(G, ρ) is dim Hom G (π ⊗ ρ, τ ).In general, this is a directed graph.
We will consider undirected graphs as a special case of directed graphs.If ρ is selfdual then N π,τ = N τ,π and the adjacency matrix of the graph Γ(G, ρ) is symmetric.In this case we will consider the McKay graph as an undirected graph, with an undirected edge corresponding to a pair of directed edges of opposite directions.1.2.Main results.We work over the base field C. In this paper we consider McKay graphs which are (undirected) forests: these are undirected graphs without closed paths (in a path, no repetitions of edges are allowed).In particular, in such a graph there is at most one edge between each 2 vertices, and no edges from a vertex to itself.
Theorem 1.1 (See Proposition 5.1, Theorem 5.2).Let G be a finite group and ρ its representation.
Assume that the McKay graph Γ(G, ρ) is a forest.Then we have: (1) ρ is irreducible and self-dual.
(2) Each of these components is isomorphic to one of the following: (c) A "hedgehog": a tree with 4 n leaves and one vertex connected to all of these leaves (1) , where n ⩾ 0. Furthermore, if one of the components is a "hedgehog" with 4 n leaves, n ̸ = 1, then all the remaining components are hedgehogs with the same number of leaves, 4 n .
Our second result concerns McKay graphs which are trees (i.e.connected forest graphs): Theorem 1.2 (See Theorem 4.1).Assume Γ(G, ρ) is a tree.Then the pair (G, ρ) is isomorphic to one of the following pairs: (1) G is a dihedral group of order divisible by 4, and ρ is the complexification of its natural 2-dimensional real representation.this graph is the union of n copies of the tree Γ(G, ρ).
Constructing and classifying forests whose connected components have different sizes seems to be a much more challenging problem.We give several examples of such forests in Section 5.2.1.3.Background and motivation.Let G be a finite group.Then one can reconstruct the isomorphism class of G from the monoidal category of representations of G. Therefore, while classifying such monoidal categories is a central and interesting problem, in full generality it is as complex as the problem of classifying finite groups.A numerical shadow of the monoidal category of representations of G is the character table of G.It is well-known that this shadow is not enough to reconstruct the group G (for instance, the dihedral group Dih 4 and the quaternion group Q 8 have the same character table).Yet even the question of describing all possible character tables still looks very difficult in general.The McKay graph is a combinatorial way to describe a piece of this character table.In fact, the data in the character table of a group G is equivalent to the data of all the McKay graphs Γ(G, ρ) where ρ runs over the irreducible representations of G and the labeling of the vertices in the graphs is fixed.Another motivation to consider McKay graphs is the fact that they are spectral analogues of the Cayley graphs.
So the starting point for this paper is the following: ( 1) For n = 0, we obtain a graph with 2 vertices and an unoriented edge between them.It is not clear whether any of these questions can be answered in full generality.We do not expect a comprehensive answer to both of them, since this is equivalent to describing all finite groups and their representations.However, we believe that studying special cases of this problem is illuminating.
In the present paper, we give a partial answer to the first question in the case when Γ is an (undirected) forest and answer both questions in the case when Γ is an (undirected) tree.
1.4.Related works.In [4], McKay classified connected McKay graphs with spectral radius 2; this leads to the ADE classification of the finite subgroups of SU (2), and can be adapted to the classification of pairs (G, ρ) where ρ is a 2-dimensional self-dual faithful representation of G (see Proposition 3.4).This result was preceded by a classification of all connected simple graphs with spectral radius 2, due to Smith (see [5]), which were shown to be affine Dynkin diagrams of type A, D or E.
1.5.1.The case of a tree graph.One of the main methods to study a McKay graph Γ(G, ρ) is to observe that if we consider its adjacency matrix as an operator from the linear span of Irr(G) to itself, then this operator can be realized as the multiplication by χ ρ when we identify the linear span of Irr(G) with the space of class functions on G.
Recall that a graph Γ is a tree if and only if the following holds: (a) it is undirected, (b) it is connected, (c) the number of its vertices exceeds by 1 the number of its (undirected) edges.
If Γ = Γ(G, ρ) is a McKay graph, then the first condition is equivalent to the fact that ρ is self-dual, and the second condition is equivalent to the fact that ρ is faithful.So it is left to exploit the third condition.In order to do this, we prove (in Corollary 3.6) the following formula for a general undirected McKay graph Γ = Γ(G, ρ):

Here
• A Γ is the adjacency matrix of the graph Γ.
• x ranges over (a set of representatives of) the conjugacy classes of G.
• C(x) denotes the centralizer of x.We use this formula in order to deduce that if Γ is a forest then ρ is irreducible and for any non-central x ∈ G, we have (see the proof of Proposition 5.1): We finish the argument in the proof of Theorem 1.2 by considering the following cases: (a) C(x) is abelian for some x ∈ G.In this case it is easy to see that ρ is 2dimensional, so we can use McKay's classification (see Section 1.4).
Algebraic Combinatorics, Vol. 6 #2 (2023) (b) C(x) is non-abelian for any x ∈ G.In this case we prove that G is an extra special 2-group (i.e. a 2-group of unipotent depth 2 and center of order 2).Since such groups are classified (see [1,3]), this completes the argument behind the proof of Theorem 1.2.
1.5.2.The case of forests.We give below an overview of the main ingredients in the proof of Theorem 1.1.(3) we use Smith's classification of graphs with spectral radius 2 (see [5]) to state that all the connected components will be affine Dynkin diagrams of types D or E. (e) If the principal component is a hedgehog, we use (3) to show that |T | = 1 for any T ∈ Irr(N )//G, and conclude that all the connected components in this case are isomorphic to the principal one.
We then give some examples of McKay graphs which are forests with nonisomorphic connected components.
It would be interesting to understand which combinations of affine Dynkin diagrams of types D, Ẽ may occur, and for which groups G the disjoint unions of hedgehogs may appear.
In a follow-up paper, we plan to investigate this question further.
1.6.Structure of the paper.In Section 2 we give the required preliminaries on groups and McKay graphs, including the definition of the extraspecial groups (see Section 2.2) and the McKay correspondence between affine Dynkin diagrams and finite subgroups of SU (2) (see Section 3.1).In Section 3 we prove some auxiliary results, such as a criterion for a McKay graph to be connected bipartite and the formula which we use later on.In Section 4 we prove Theorem 1.2.In Section 5 we prove Theorem 1.1 and construct some non-trivial examples of forests in Subsection 5.2.

Preliminaries and notation
The base field throughout the paper is C. • We denote by 1 G its unit element, by Z(G) its center, by G//G the set of conjugacy classes of G, and by C(g) the centralizer of g ∈ G.
• We denote by Irr(G) the set of its irreducible representations and by 1 its trivial representation.• We denote by C[G] adj the conjugation representation of G on its group algebra.• For any representation ρ of G, we denote by χ ρ : G → C its character, where χ ρ (g) := tr(ρ(g)).By abuse of notation, we use the same notation for the corresponding function • We denote by X(Γ) the set of vertices of Γ.
• We denote by N(x) the multiset of neighbors of a vertex x in Γ: those are y ∈ X(Γ) with an edge x → y in Γ (if there are several edges x → y, then y appears with appropriate multiplicity in N(x).• A loop is an edge connecting a vertex to itself).
• A graph Γ is called simply-laced if for each x, y ∈ X(Γ) it contains at most one edge x → y. • A walk in Γ is a sequence of edges (e 0 , . . ., e s ) such that the end of e i is the beginning of e i+1 for each i < s (repetitions of edges are allowed!).A walk is called a circuit if the end of e s is the beginning of e 0 .A path is a walk without repetitions of edges.• Given a pair of edges in opposite directions (or a loop), we will usually draw a single undirected edge instead.If all the edges in the graph are undirected, we will say that this graph is undirected.• We say that Γ is an undirected tree if Γ is undirected and when considered as an undirected graph, it contains no circuits (in particular, it is simply-laced and without loops!).
Notation 2.5.Given a graph Γ with vertex set X, we denote by A Γ the adjacency matrix of this graph.That is, A Γ is an |X| × |X| matrix with integer coefficients whose rows and columns are indexed by the set X.The (x, y) ∈ X × X entry of A Γ is the number of edges in Γ from x to y.The matrix A Γ naturally defines a linear endomorphism of the space of functions X → C, and we will identify A Γ with this endomorphism.
Another (non-isomorphic) extra special group of order 8 is the quaternion group Q 8 .• Given a finite-dimensional vector space V over the field F p , we can define the generalized Heisenberg group Heis(V ) as follows: the underlying set is V × V * × F p and the multiplication is given by where − • − denotes the obvious pairing V * × V → F p .This group Heis(V ) is an extra special p-group of order p 1+2 dim V .
It turns out that such groups exist only if N is even, i.e.N = 2n for some n ⩾ 0; moreover, for each power p 1+2n (n ⩾ 1) there exist precisely two isomorphism classes of extra special p-groups of order p 1+2n .
For p > 2, any extra special p-group of order at least p 3 is a central product of several copies of the two groups of order p 3  An extra special p-group G of order p 1+2n has precisely p 2n representations of dimension 1 on which the center Z(G) acts trivially, and p − 1 representations of dimension p n on which the center Z(G) acts by a non-trivial character.Here the vertex in the center corresponds to τ .This graph coincides with the McKay graph for (Q 8 , µ) where Q 8 is the quaternion group and µ is the obvious 2-dimensional (irreducible) representation given by H ∼ = C 2 .This coincidence is not surprising, given that the character tables of Dih 4 , Q 8 coincide.
Example 2.11.Consider an extra-special 2-group G of order 2 1+2n , n ⩾ 0 as described in Section 2.2 (for n = 0, the corresponding extra special group is unique, and it is just C 2 ).Then G has 2 2n + 1 irreducible representations: 2 2n irreducible representations of dimension 1 coming from G/Z(G) ∼ = F 2n 2 and one irreducible representation ρ of dimension 2 n .The McKay graph for (G, ρ) is an unoriented connected graph of th following form: it has one vertex ρ with 4 n unoriented edges coming out, and 4 n vertices connected only to ρ by a single unoriented edge.We will call such a graph a hedgehog with 4 n spines.Below is an example for n = 2: The following statement is well-known (see [4, Proposition 2]): Lemma 2.12.Let ρ be a representation of the group G.A corresponding eigenvector for the eigenvalue χ ρ (g) is χ(g) : Irr(G) → C, µ → χ µ (g) (a column of the character table ).
Since χ ρ (g), g ∈ G is the sum of eigenvalues of ρ(g), all of which are roots of unity, we have:

The eigenvalue of A Γ(G,ρ) with the maximal absolute value (the spectral radius of
The spectral radius of a real matrix A with non-negative entries has an important property given by the Frobenius-Perron theorem: the corresponding eigenspace of A contains a vector all of whose entries are non-negative. In fact, if A is an "irreducible" matrix (one which cannot be presented as a block matrix with non-trivial blocks), the spectral radius has multiplicity 1 as an eigenvalue of A, and it is the only eigenvalue of A for which an eigenvector with positive entries exists.For example, the adjacency matrix of a strongly connected graph Γ (one where there is a directed walk from each vertex to any other vertex) is always irreducible, and hence its spectral radius has multiplicity 1.

Shapes of McKay graphs. Let us give some basic examples of graphs which can or cannot appear as McKay graphs Γ(G, ρ) for some G, ρ.
Example 2.15.
(1) Given a group G and a representation ρ of G, the representation ρ is 1-dimensional if and only if the McKay graph Γ(G, ρ) is a disjoint union of directed cycles (some of the cycles might consist of 1 vertex only, with a single loop).Yet a given a complete (undirected, loop-less) graph on n vertices, we can always find a McKay graph Γ(G, ρ) containing our complete graph as a subgraph: for instance, take G = S N (the symmetric group) for N >> n and ρ to be the reflection representation.

Connected components in McKay graphs. Given two vertices µ, λ in a
McKay graph, we can ask whether there is a directed walk from µ to λ, or whether there is a walk from µ to λ if we forget about the directions of the edges in our graph.Although for general graphs these relations do not coincide, it turns out that in McKay graphs they do (see [2, Section 3]), so the notion of a connected component in a McKay graph is intuitive; each such component is "strongly connected".
Given a McKay graph Γ(G, ρ), the connected component Γ 1 containing the vertex The following statement is well-known, see for example [4, Proposition 1], [2, Proposition 3.3]: Lemma 2.16.Let ρ be a representation of the group G. Then ρ is faithful if and only if Γ(G, ρ) is connected (that is, for any µ, λ ∈ Irr(G) there exists a walk from µ to λ in Γ(G, ρ)).
In fact, given any representation ρ of G with kernel N = Ker(ρ), the principal component Γ 1 is isomorphic to Γ(G/N, ρ).The connected components of Γ(G, ρ) correspond to blocks in the adjacency matrix A Γ(G,ρ) ; the number of such blocks is then the multiplicity of the largest eigenvalue, implying the following lemma (see also [2, Proposition 3.10, Lemma 4.1]): Lemma 2.17.
(1) The number of connected components of Γ(G, ρ) is precisely the number of G-conjugacy classes of N ; the latter equals the number of G-orbits in Irr(N ) (G acts on Irr(N ) twisting the action of N on the representations).
In fact, there is a bijection given as follows: to every connected component The group G of characters G → C × acts on the graph Γ(G, ρ) by tensoring each vertex with the appropriate 1-dimensional representation.
Algebraic Combinatorics, Vol. 6 #2 (2023) Proof.Since ρ is self dual, it either has a symplectic G-invariant form (i.e.ρ is of quaternionic type) or a symmetric G-invariant form (i.e.ρ is of real type).In the former case, G ⊂ SL(2) ∩ U (2) = SU (2) and we are in the situation of Theorem 3.3.
In the latter case, the representation (ρ, V ) is of real type, the corresponding 2dimensional real representation (ρ R , V R ) satisfies: Proof.The eigenvalues of the matrix A Γ are precisely χ ρ (g) for g ∈ G//G (see Proposition 2.13), so the sum of eigenvalues of A k Γ is g∈G//G χ ρ (g) k , which proves the middle equality.The right equality holds for any graph Γ and is obvious.
It remains to prove that dim Hom In the right hand side, each summand dim Hom G (ρ ⊗k ⊗ µ, µ) is precisely the diagonal entry in A k Γ corresponding to µ ∈ Irr(G), so as required.Proof.Let X(Γ) = X 0 ⊔ X 1 be the bipartition of the set of vertices, such that ∀i ∈ {1, 2}, ∀x, y ∈ X i , x / ∈ N(y).
Let f : X(Γ) → C be the eigenvector of A Γ with eigenvalue λ.Then Then for any x ∈ X i , we have: Proof.[⇐]: Assume C 2 ⊂ Z(G), and let z ∈ C 2 ⊂ Z(G), z ̸ = 1 G .Then the vertices in Γ can be partitioned into two disjoint subsets, according to the eigenvalue by which z acts on the given irreducible representation of G. Since ρ is faithful and irreducible, ρ(z) = − Id.So clearly, there are no edges in Γ between vertices λ, µ ∈ Irr(G) such that λ(z) = µ(z).Hence Γ is bipartite.
Next, the eigenvalues of A Γ are precisely χ ρ (g) by Proposition 2.13, so there exists Since z ∈ G has finite order, ρ(z) is a diagonalizable operator whose eigenvalues are all roots of unity.We have tr(ρ(z)) = − dim ρ and hence ρ(z) = − Id.Faithfulness of ρ now implies that z ∈ Z(G) and z 2 = 1 G .Hence C 2 ⊂ Z(G) as required.□ Lemma 3.9.Assume ρ is irreducible and faithful, and that ρ Proof.By Schur's lemma, ρ defines a character of the center where ρ(z) = λ(z) Id.Since ρ * ∼ = ρ, we have: 1 ⊂ ρ ⊗ ρ * ∼ = ρ ⊗2 and on this space z ∈ Z(G) acts by λ 2 (z) Id.So λ(z) 2 = 1 and thus λ(z) = ±1.Now, since ρ is faithful, the map λ : Since we assumed that the group C(g) is itself not abelian, we can find h ∈ C(g) which is not aligned with g.We have: h / ∈ Z(G), so we denote by t 1 ̸ = t 2 the distinct eigenvalues of ρ(h).Since h is not aligned with g, the eigenvalues of ρ(gh) would be {t j z k |k, j ∈ {1, 2}} or at least 3 of these products.Since ρ(gh) also has at most 2 distinct eigenvalues, this implies that in at least one of the pairs (t 1 z 1 , t 2 z 2 ), (t 1 z 2 , t 2 z 1 ) the elements coincide.Without loss of generality, let us assume that t 1 z 1 = t 2 z 2 = t1 z1 .So the eigenvalues of ρ(gh) are either {t 1 z 1 , t 1 z 2 } or {t 1 z 1 , t 2 z 1 }.Then So t 1 z 1 = ±1 (implying t 1 = ±z 2 ) and the eigenvalues of ρ(gh) must be ±1.Thus at least one of the following equalities holds: z 2 2 = ±t 1 z 2 = ±1 or z 2 1 = ±t 2 z 1 = ±1, any of which implies z 1 , z 2 ∈ {±i}.So we see that for every g ∈ G, the eigenvalues of ρ(g) are either {±i} or a subset of {±1}.
In particular, ρ(g) 2 = ± Id for each g ∈ G, so G/Z(G) is a group whose elements all square to identity.Hence G/Z(G) is of the form F m 2 (an elementary abelian 2group).Since Z(G) = C 2 , we conclude that G is an extra special 2-group, m is even and ρ is the unique irreducible representation of G of dimension greater than 1 (see Example 2.11).This concludes the proof of the theorem.□

Forests
In this section we investigate the case of an undirected forest graph, i.e. a disjoint union of undirected trees.

( 2 )
G ⊂ SU (2) is a binary polyhedral group of type D or E and ρ is the natural 2-dimensional representation of G. (3) G is an extra special group of order 2 1+2n for some n ⩾ 0, i.e. a central extension of (Z/2Z) 2n by Z/2Z.In that case, ρ is the unique irreducible representation of G of dimension 2 n .Remark 1.3.The connected component of the McKay graph Γ(G, ρ) containing the trivial representation is called the principal component of Γ(G, ρ) and it is itself a McKay graph for the group G/ Ker(ρ) and its faithful representation ρ.If Γ(G, ρ) is a forest, our second result describes the pair (G/ Ker(ρ), ρ) up to isomorphism.Finally, we construct some examples of McKay graphs which are forests with nonisomorphic connected components.Constructing such forests where all the connected components are isomorphic to a given tree from the list above is very easy: for n ⩾ 2, let (G, ρ) be as in Theorem 1.2, and consider the McKay graph Γ(G × Z/nZ, ρ ⊠ 1); (a) Let N = Ker(ρ) and consider the action of G on Irr(N ) induced by the conjugation action of G on N .It is well-known that the set of connected components of a McKay graph Γ(G, ρ) is naturally indexed by the set of Gorbits Irr(N )//G.The indexing is obtained by restricting a representation µ ∈ Γ(G, ρ) to N (see Lemma 2.17).(b) Let τ ∈ Irr(N ) and let T ∈ Irr(N )//G be the G-orbit of τ .Let Γ T (G, ρ) be the connected component of Γ(G, ρ) corresponding to T .Then we have (see Lemma 2.19):
Theorem 3.3  relies on the classification of all finite undirected connected simply-laced graphs Γ with spectral radius 2. The classification of such graphs states that they must be affine Dynkin diagrams of types A, D or E, which was proved by Smith in[5] (cf.[4, Proposition 4]).Proposition 3.4.Let G be a finite group and ρ be its self-dual faithful representation (so Γ(G, ρ) is connected and undirected).Assume dim ρ = 2. Algebraic Combinatorics, Vol. 6 #2 (2023) Then either G ⊂ SU (2) (hence G appears in the list of Theorem 3.3 and Γ(G, ρ) is of the form above), or G = Dih n and ρ is the complexification of the tautological 2-dimensional representation of Dih n .In the latter case, Γ(G, ρ) is drawn below.parity of n McKay graph for Dihn, tautological ρ, dim(ρ) = 2 number of vertices

3 . 2 .Lemma 3 . 5 .
where ρ R is a faithful 2-dimensional representation of G over R. Hence G ⊂ Dih n for some n ⩾ 1.But in that case G it either cyclic (and then we are again in the situation of Theorem 3.3) or G is dihedral.This completes the proof of the proposition.□ McKay graphs and characters.Let Γ := Γ(G, ρ) be the McKay graph for a finite group G and a representation ρ of G. Let k ⩾ 1.We have: , making f an eigenvector of A Γ with eigenvalue −λ.□ Let Γ := Γ(G, ρ) be the McKay graph for a finite group G and a representation ρ of G. Lemma 3.8.Assume ρ is irreducible and faithful.Then the graph Γ = Γ(G, ρ) is bipartite if and only if C 2 ⊂ Z(G).

5. 1 .Proposition 5 . 1 . 1 ) 2 )Remark 5 . 3 . 4 . 5 . 4 .
Classification of McKay forests.Let Γ := Γ(G, ρ) be the McKay graph for a finite group G and an representation ρ of G. Assume Γ is an (undirected) forest.Then ρ is a irreducible, self-dual representation.Proof.Let N := Ker(ρ) ◁ G and let K be the number of connected components of Γ.By Lemma 2.17, we have: K is the number of G-conjugacy classes in N .The principal component Γ 1 = Γ(G/N, ρ) is a tree, so Theorem 4.1 shows that ρ is a self-dual irreducible representation of G/N .Hence ρ possesses the same properties when considered as a G-representation, which proves the first part of the statement.□ Theorem 5.2.Let Γ := Γ(G, ρ) be the McKay graph for a finite group G and a representation ρ of G. Let N := Ker(ρ).Assume Γ is an undirected forest.Then we have: (All the connected components of Γ are of one of the forms described in Theorem 4.1.(If one of the components is a hedgehog with 4 n spines for n ̸ = 1, then all the components of Γ are of isomorphic to each other (in particular, they are isomorphic hedgehogs).For n = 1, the hedgehog with 4 spines is the affine Dynkin diagram D Remark In particular, if one of the components is an affine Dynkin diagram of types D of E, the remaining components are also affine Dynkin diagrams of types D or E. They do not have to be isomorphic to each other, as we will show in the examples of Section 5.2.
Notation 2.2.For n ⩾ 2, we will denote by C n the cyclic group of order n.For prime p, the elementary abelian group p-group C ×k representation ρ and an H-representation τ , we denote by ρ ↓ H the restriction of ρ to H and by τ ↑ G H the induction of τ to G. p will sometimes be denoted F k p .Notation 2.3.For n ⩾ 3, we will denote by Dih n the dihedral group of order 2n (group of symmetries of the regular n-gon).We will also denote by Dih 2 the group C 2 × C 2 (considered as the group of symmetries of a digon).Notation 2.4.Let Γ be a directed graph.
[3]liminaries on extra special groups.A special class of finite groups which appear in this paper are extra special groups.For an introduction to extra special groups, see[1]and the appendix of[3].There is just one extra special group of order p, up to isomorphism, which is C p .• There are two examples of extra special p-groups of order p 3 : the Heisenberg group Heis(p) given by upper-triangular 3 × 3 matrices over the field F p with 1's on the diagonal, and the semidirect product C p 2 ⋊ C p where C p acts nontrivially on C p 2 .These are non-isomorphic for p > 2. • For p = 2 the above examples coincide: we have Heis(2) 6.A graph Γ is undirected if and only if A Γ is symmetric.Algebraic Combinatorics, Vol. 6 #2 (2023) 2.2.Definition 2.7.Let p be a prime number.A finite group G is called an extra special p-group if Z(G) ∼ = C p and G/Z(G) ∼ = (C p ) N = F Np for some N ⩾ 0.
appearing in Example 2.8.If all of the n factors appearing in the central product are isomorphic to Heis(p) then the obtained group is generalized Heisenberg group Heis(F n p ) and it is p-torsion.If at least one of the n factors is of the form C p 2 ⋊ C p then we obtain another, non-isomorphic extra special group of order p 1+2n , but now having some elements of order p 2 .For p = 2, a similar situation occurs: any extra special 2-group of order at least 8 is a central product of several copies of Dih 4 and Q 8 .If all of the n factors appearing in the central product are isomorphic to Dih 4 then the obtained group is the generalized Heisenberg group Heis(F n 2 ).If at least one of the n factors is of the form Q 8 then we obtain another, non-isomorphic extra special group of order 2 1+2n .
Any disjoint union of directed cycles of equal size can appear as a McKay graph: if we have k disjoint directed cycles of size n, this is the McKay graph for (Z/nZ×G, τ ⊠1) where τ is an irreducible representation of Z/nZ, and G is any finite group with precisely k irreducible representations (e.g.G = Z/kZ).(2) A McKay graph Γ(G, ρ) cannot be a directed tree or a disjoint union of such: indeed, one cannot have "leaves" (i.e.vertices with only one edge coming in or going out) in a directed McKay graph.(3) Considering the opposite extreme situation, a complete (undirected, loop-less) graph on n vertices -such a graph can be obtained as Γ(Z/nZ, ρ).Here ρ is the representation of Z/nZ on C n /Span{(1, . . ., 1)}, where Z/nZ acts on C n by shifting cyclically the coordinates.(4) A complete graph on n vertices (n > 2) cannot be obtained as a McKay graph Γ(G, ρ) if we require ρ ∈ Irr(G).This is due to the fact that in any McKay graph Γ(G, ρ) with ρ ∈ Irr(G), the vertex 1 has only one neighbor: the vertex ρ.
□Corollary 3.6.Assume Γ is undirected and loop-less.Then dim Hom G (ρ ⊗2 , C[G] adj ) = Let Γ be a bipartite graph with adjacency matrix A Γ .Then for each eigenvalue λ of A Γ , −λ is also an eigenvalue of A Γ .