Complement Graph 81
Figure 4.3: This is the disconnected graphU(3,3)on 9 vertices.
Remark 4.3.1. Recall from Section 2.3.4, both graphs are the two non-trivial strongly regular graphs with exactly2eigenvalues.
Lemma 4.3.2. The graph U(n, r) has singular endomorphisms of ranks r,2r,3r, ...,
(n−1)rand the number of endomorphisms of rankk·ris given by
n k
·S(n, k)·k!·(r!)n,
whereS(n, k)is the Stirling number of the 2nd kind (counting the number of partitions ofnelements intokparts).
Proof. The ranks of the endomorphisms are r,2r,3r, ...,(n −1)r, since the complete graphs form maximal cliques. Now, we count the endomorphisms of rankk·r. Letφbe such an endomorphism. The image ofφis a union ofkgraphsKr. Thus, choosekout of
thenfactors. The kernel classes ofφare the parts of a partition ofncomplete graphs into
kparts. Each kernel class is mapped to one of thek complete graphs in the image ofφ, providingk!choices. Finally, note that each complete graphKr is mapped to a complete
graphKrinr!ways.
Unfortunately, it is much harder to describe the complementary graph T(n, r) =
U(n, r), as many more choices arise. One can easily deduce that the rank of a singular endomorphism could be any number between n and r · n −1. In the next result we consider the simple caseT(n,2)and postpone the general case until Lemma 4.3.6.
82 Endomorphisms
Lemma 4.3.3. The number of singular endomorphisms ofT(n,2)is (2n−1)·2n·n!.
Proof. This graph is multi-partite with 2 vertices in each part. Since φ is a singular endomorphism, φ collapses at least one part to a single vertex. An endomorphism of rankn+k collapsesn−k parts; hence, there are nchoose k choices to pick the parts which are collapsed. Moreover,φ maps a part to a part, providing n! choices. F, since a part is mapped to a part, there are two choices for the two points to be mapped to. In total, this gives2n choices. Finally, summing over all ranks and not counting full ranks,
that is endomorphisms of rank2n, we obtain the result.
Generators ofEnd(Γ)
The automorphism group of these graphs is the wreath productSroSnwith the imprimi-
tive action. This rather large automorphism group covers a lot of symmetry leading to a very small relative rank. In fact, the relative rank is1for both graphs.
Lemma 4.3.4. 1. The monoidEnd(U(n, r))has singular rank1and its singular gen- erator is given byt, wheretis collapsing two of the componentsKr and fixing the other components pointwise.
2. The monoidEnd(T(n, r))has singular rank 1and its singular generator is given byt, wheretis collapsing two points in one of the parts and fixing all other points.
Example 4.3.5. ForU(3,3)from Figure 4.3 a generating transformation ist1, where
t1 = 1 2 3 4 5 6 7 8 9 1 2 3 1 2 3 7 8 9 .
4.3. The Complete Multi-Partite Graph and its
Complement Graph 83
ForT(3,3)a generating transformation ist2, where
t2 = 1 2 3 4 5 6 7 8 9 1 1 3 4 5 6 7 8 9 .
Given the generator forEnd(T(n, r)), we can determine the number of singular endo- morphisms quite effortlessly with the help of a computer. Taking the additional symmetry into account we get.
Lemma 4.3.6. The number of singular endomorphisms ofT(n, r)is
r(r−1)n−((r−1)!)n·rn·n!.
Proof. This formula follows when taking the symmetry into account which arises for
r > 2. The both factors on the right hand side correspond to the two factors in Lemma 4.3.3. Only the factor r(r−1)n−((r−1)!)n
is somewhat more complicated. However, this factor comes from combining the different kernel types. We determine the number of singular endomorphisms for each possible rank individually and then sum over all ranks. For each rank there might be several kernel types which can occur and the result follows from applying binomial identities.
Structure ofEnd(Γ)
As we have seen by determining the endomorphisms of T(n, r) and U(n, r), the en- domorphisms ofT(n, r) are wilder and less structured; whereas the endomorphisms of
U(n, r)are straightforward to describe. For this, Green’s relations inEnd(U(n, r))be- have much better.
So in this section, we determine the number ofD-,H-,L- andR-classes and the struc- ture of the H-classes in End(U(n, r)). However, we will not be able to do this for
End(T(n, r))because the difficulties mentioned above, except for the caser = 2which will be deferred to the following section.
84 Endomorphisms
As the transformations inU(n, r)have an obvious structure we can deduce.
Lemma 4.3.7. LetΓbe the graphU(n, r), forr, n≥2. Then,End(Γ)is regular.
Proof. Elements in End(Γ) permute the elements within the subgraphs Kr and they
permute and collapse the blocks Kr. Thus, for any a ∈ Sing(Γ) there is an element
g ∈Aut(Γ)such thatagis an idempotent. Now, by the identity
ag= (ag)2 =agag ⇔a=aga
the elementais regular.
The following basic result is key in determining theD-classes.
Proposition 4.3.8(Prop 3.6, [57]). LetSbe a subsemigroup of a semigroupT, letDxbe a regularD-class ofSandya regular element ofS.
1. xandyare in the sameL-class withinT if and only if they are in the sameL-class withinS.
2. xandyare in the sameR-class withinT if and only if they are in the sameR-class withinS.
The structure of the endomorphisms leads to a simple way to distinguishD-classes, but alsoLandR-classes. It holds the same as for the full transformation monoidTn.
Lemma 4.3.9. Fora, b∈End(Γ)it follows.
• a L b⇔im(a) = im(b).
• a R b ⇔ker(a) = ker(b).
• a D b⇔rank(a) = rank(b).
Proof. Two elementsaandb inTnare in the sameL-class if they have the same image
[57, Lemma 3.1]. Similarly, they are in the same R-class if they have the same kernel. So the first two statements follow from the previous lemma.
4.3. The Complete Multi-Partite Graph and its
Complement Graph 85
Next, assumeaandbare in the sameD-class. Then, there is an elementcwhich is in the sameR-class asband in the sameL-class asa. This implies thatrank(a) = rank(b). On the other hand, if the rank ofaandbis the same then we can easily find an element
g ∈ Aut(Γ) such that both a andbg have the same image andb and bg have the same kernel. But thenaandbneed to be in the sameD-class.
Corollary 4.3.10. End(Γ) has n D-classesD1, ..., Dn, where Dk contains all the ele- ments of rankk·r.
Next, we count theL, RandH-classes in eachD-class.
Lemma 4.3.11. Let Dk the D-class containing exclusively elements of rank k · r, for
1≤k < n. Then, theD-classDk has
• nk
L-classes,
• (r!)n−kS(n, k)R-classes, whereS(n, k)is the Stirling number of2nd kind,
• nk
(r!)n−kS(n, k)H-classes, each containingk!(r!)k elements, and,
• nk
(r!)n−kkn−kH-classes contain an idempotent and, thus, are groups.
Proof. This follows from simple counting arguments, where for the last part this is also equal to the number of idempotents of rankk·r.
Proposition 4.3.12. Let Dk be theD-class with elements of rank k·r, for1 ≤ k < n. Then, the subsemigrouphDkihas the following structure
hDki=Dk]Dk−1]Dk−2] · · · ]D1.
Proof. This can be immediately seen from the action ofSing(Γ)on the graph.
AsD1is the bottomD-class, this is the minimal ideal of the semigroup and thus it is
86 Endomorphisms
Proposition 4.3.13. Let H be anH-class containing an idempotent in theD-classDk, for somek = 1, ..., n. ThenHhas the structure
H∼=SroSk,
where the action ofSroSkis the imprimitive wreath product action.
Proof. Since kernel and image are determined by theLandR-class, the elements inH
can merely permute the blocksKrwithin the image and permute the elements withinKr
itself.
Corollary 4.3.14. TheH-classes inD1are isomorphic to the symmetric groupSr.
A final result considers the regularity of both graphs.
Proposition 4.3.15. The endomorphism monoids ofU(n, r)andT(n, r)are regular, for allnandr.
Proof. It is left to verify thatSing(T(n, r))is regular. However, this follows from McAl- ister’s result [66, Thm. 3.10].
The Cocktail Party Graph and the Ladder Graph
The graphs U(n, r) and T(n, r) have other popular names for r = 2; U(n,2) is also called the ladder graphLD(n) (or ladder rung graph), whereas T(n,2) is the cocktail party graphCP(n). These two graphs appear on various occasions, and in particularly
CP(n) constitutes one of the three families of strongly regular graphs with minimal eigenvalue−2.
Here, we focus onCP(n)asLD(n)was implicitly covered in the previous section.
Lemma 4.3.16. Let Γ be CP(n) and let End(Γ) be its endomorphism monoid. Then Lemma 4.3.9 is valid.
4.3. The Complete Multi-Partite Graph and its Complement Graph 87 k= 0 k = 1 k = 2 k = 3 k = 4 k = 5 k = 6 n =1 1 2 2 1 4 4 3 1 6 12 8 4 1 8 24 32 16 5 1 10 40 80 80 32 6 1 12 60 160 240 192 64 7 1 14 84 280 560 672 448 8 1 16 112 448 1120 1792 1792 Table 4.2: #L-classes inDkof the graphCP(n)forr= 2.
Proof. Because we haver = 2 that is because we considerCP(n), the proof is exactly the same.
Example 4.3.17. The last point of Lemma 4.3.9 is not true for r > 2, in general. For instance it is not true forT(3,3)because the two transformations [1,1,1,4,5,6,7,8,9]
and[1,1,3,4,4,6,7,8,9]do not lie in the sameD-class.
Corollary 4.3.18. Two transformationsaandbare in the sameD-class ofEnd(Γ)if and only if they have the same rank.
An additional motivation for considering the endomorphisms ofCP(n)is their con- nection to interesting number sequences. For instance, there is a correspondence between the number ofL-classes in aD-class and the numbersPn−kwhich are defined below. Ta-
ble 4.2 lists the number ofL-classes in Dk, and for each k the corresponding column
gives a subsequencePn−kwhich is interesting by itself. For further reference confer the
OEIS library [70].
Definition 4.3.19. IfX1, ..., Xnis a partition of a2n-setXinto2-blocks, then letPn−k denote the number ofk-subsets ofX containing none ofXi, fori= 1, ..., n.
The relation between the sequencePn−kand the endomorphisms should be clear from
the definition.
Lemma 4.3.20. LetDk be theD-class with elements of rankk. Then,Dkhas 2nn−k
=
n k−n
88 Endomorphisms
Endomorphisms of Strongly Regular Graphs with
Minimum Eigenvalue -2
In the previous sections we described the endomorphism monoids of the infinite families of graphs covered by Seidel’s theorem (Theorem 2.3.10) on the strongly regular graphs with minimal eigenvalue−2. Moreover, by a straightforward computation we checked that the remaining 7 graphs from this theorem do not admit singular endomorphisms. Hence, we obtain the following result.
Corollary 4.4.1. 1. The square lattice graph L2(n), for n ≥ 3, has uniform sin-
gular endomorphisms of rank n and the number of singular endomorphisms is 2n·#of Latin squares of ordern.
2. The triangular graph T(n), forn ≥ 5, has no singular endomorphisms for odd
n. But, for evenn the singular endomorphisms are uniform of rankn−1and the number of singular endomorphisms isn!·(#of1-factorisations ofKn).
3. The cocktail party graphCP(n), forn ≥2, has singular endomorphisms of ranks
n, n+ 1, n+ 2, ...,2n−1 and those are the only possible ranks. Moreover, the number of singular endomorphisms is(2n−1)·2n·n!.
4. The remaining7graphs have no singular endomorphisms, thus they are cores.
Various Grid Graphs and their Endomorphisms
Orthogonal Array Graphs
In this section, we summarise the newly established connections and results on orthogo- nal array graphs and their singular endomorphisms.