CIRCUITS AND TREES IN ORIENTED LINEAR GRAPHS

by T. van Aardenne-Ehrenfest (Dordrecht) and N. G. de Bruijn (Delft)

§ 1. P n (a)-cycles.

In this § we state the problem which gave rise to our investigations about graphs. The further contents of the paper are independent of this § 1.

Consider a set of a figures 1, 2, ... , a, and let n be a natural number. A sequence of n figures will be called an n-tuple. Clearly, there are an different n-tuples.

An oriented circular array, consisting of an figures, will be called a P,,-cycle, whenever it has the property that each n-tuple occurs exactly once as a set of n consecutive figures of the cycle. An example, with a = 3, n = 2 is the cycle 1 1 2 2 3 3 1 3 2.1)

The existence of P ,,(a)-cycles, for arbitrary values of a and n, was proved by M. H. MARTIN [3J, 1. ]. GOOD [2Jand D. REEs [4]. One of us showed ([IJ) that, for a = 2, the number of different P n(2)-cycles equals 2f(n), f (n) = 2n-l __ n.

This result was derived as follows. The number of P ,,(21-cycles can be interpreted as the number of circuits in a certain graph N n+1 (compare also [2J). The graph Nn+l can be obtained by a certain operation from N n' and by a general theorem on circuits in oriented graphs the number of circuits of Nn+1 could be expressed in the number of circuits of N". This theorem on graphs was proved in [IJ only for the case that at any vertex 2 edges point outward and 2 inward. In the present paper we shall deal, among other things, with the general case (theorem 4). This result immediately enables us to determine the number of P ,,(a)-cycles for arbitrary a. Referring to [IJ for details, we only state the result: The number of different P n(a)-cycles is ern (a !)q, where q = an-I.

For example, there are 24 P2(3l-cycles. Six of them are

1 2 3 3 2 2 1 3 12322 1 3 3 1 2 2 3 3 2 1 3

1 2 2 3 2 1 3 3 1 2 2 3 3 1 3 2 1 223 1 332

1) It has to be understood that these figures have to be placed around an oriented circle. Therefore, 21 is one of the 2 - tuples occurring in the cycle. Naturally, 112233132 and 331321122 are considered as one and the same cycle, but 112313322, which has the reversed order, is a different one.

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Another six are obtained from these by interchanging the figures 2 and 3 everywhere. By reversing the orientation, 12 new cycles arise.

§ 2. Preliminaries about permutation groups.

Let €i m be the symmetric group of degree m, that is, the group of all m! permutations of a set Em of m objects. If ~ is a subset of €i m then the number of cyclic permutations in ~ will be denoted by 1 ~ I, and the total number of elements in ~ by n(~).

A subset 'II of €i m will be called a D-set (in em), whenever it has the property that 1 5 'II 1 has the same value for all SEem. It is easily seen that, in that case, we have 1 5 'II 1 = m-I . n ('II). For, if C is any cyclic permutation, then there are exactly n ('II) possibi­lities for 5 such that S'II contains C, and it follows that m! 1 5 'II I

= (m-1)!n('II). Furthermore, it may be remarked that I 5 'II I == ! 'II 5 i, since

SBS-I is cyclic whenever B is cyclic. Therefore, if 'II is aD-set and if P is an arbitrary element of em, then 'II P is also aD-set.

em itself clearly is a D-set in em, but theorem 1 will show that non-trivial D-sets exist.

Let E, be a sub-set of the set of objects Em' containing lobjects. Consider the sub-group @ Cern of all permutations which only permute the elements of E z, leaving the remaining elements of Em invariant. If G is any permutation of @, then G denotes the corres­ponding permutation of the objects of E " that is to say, we disregard the objects belonging to Em - E z (which are invariant under G).

G is defined uniquely by G, and vice versa. The same notation will be used for sets: if ~ C @, then ill denotes the set of all G, where G E~.

L e m mal. Let \8 be a sub-set of @ such that 58 is aD-set in @, and let C Eem be a cyclic permutation. Then we have

1\8 C I = l-I . n (\8).

Proof. We shall deal with the cyclic representations of the

permutations involved. Let G be the element of @ whose cyclic representation is obtained by cancelling the objects of Em - E I from the cyclic representation of C. Further, let GI be an arbitrary permutation of @. Then it is easily verified that GIG (of degree l) shows the same number of cycles as GIC (of degree m). Hence GIC is cyclic whenever GJ; is cyclic. Therefore

1\8 C I = 158 G I = l-I . n (58) = l-1 . n (\8).

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L em m a 2. Let m be a subset of @ such that \8 is a D-set in &. Let Q be any arbitrary permutation of Sm. Then we have

ImQI n(m)

I@QI n (@) .

Proof. If there is no G e @ such that GQ is cyclic, then both sides are equal to zero. Now assume that G e @ is such that GQ is cyclic; put GQ = C.

We have @ Q = @ C, and m Q = (m G-l) C . The set ~1 =

is (;-1 is a D-set in @, since m was a D-set in @. Now, by lemma I,

1 m Q 1 = 1 m1c 1 = 1-1 . n (m1) = 1-1 • n (m). Analogously

1 @ Q 1 = 1 @ C 1 = 1-1 • n (@), and lemma 2 has been proved.

Let k and n be natural numbers, and take m = kn. We consider a set Em of m objects, divided into k systems, each of them con­taining n objects. We shall again denote by elm the group of all permutations of Em. ~ denotes the group consisting of all k! (n!) k

permutations H with the property that Ha and Hb belong to the same system whenever a and b belong to the same system. Or, shortly, H transforms systems into systems.

The 0 rem I. ~ is a D-set in elm.

Proof. If either k = 1 or n = I, then we have ~ =elm, and the theorem is trivial.

Next we shall deal with the case k = 2, m = 2n. It has to be shown that 1 5 ~ 1 does not depend on 5 (5 eel2n). Let ~1 be the set of all permutations mapping the first system onto itself, and let ~2 be the set of those mapping the first system onto the second. Thus ~ = ~1 + ~2'

Let p be the number of objects of the first system mapped into the first system by 5, then there are q = n - p objects of the first system which are mapped into the second system. Then we have

1 5 ~11 = q{(n-l)!}2.

This can be seen, for instance, by interpreting 1 5 ~1 1 as the number of circuits (see § 3) in the following graph. Take two vertices, A and B, and 2n oriented edges: p of them from A to A, p from B to B, q from A to Band q from B to A. The number of circuits can be shown to be q {(n - I) !}2. It can be very rapidly

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determined by theorem 6, for there are exactly q trees with root A. $)2 can be written as S o ~l' where S o is an arbitrary element of

i92' Now I 5 ~21 = I 51 ~l I ' where 51 = 550, 5 1 has the same nature as S, apart from the fact that p and q changed their roles.

Hence I S$') , I~ p{(n - I )I}', and so

IS $') I ~ IS $') , 1 + IS $'), 1 ~ IP + q){ (n - l )!}'~ (n!) ' /n.

This does not depend on 5, and so our theorem has been proved in the case k = 2.

Next we consider the general case k> 2. \Ve have to show that i 51 ~ I = I S 2 ~ I for any pair 5 1. S 2 (51 e.Sm> S 2 e.Sm ) . Since any S ESm can be written as a product of transposit ions, it is sufficient to prove that I 5 .\,1 1 = 1ST ~ I for a ll 5 and for any transpositi.on T. Or, what is t he same thing, that

(2.1 )

for any Q eGrn and any transposition T fGrn .

vVe may assume that T interchanges two symbols belonging to the first and to the second system, respectively (if T inter­changes t\vo symbols of the same system, then we have $) = T $), and (2 .1 ) is t rivial). Let $)* be the sub-group of .S) consisting of all permutations of .S) which leave all individual elements of the 3rd , 4th, . .. , k/h system in variant, and le t ® be the group arising from 6 m in the same manner. Vrle now apply lemma 2, with l = 2n and m = ,!Q *. Since the theorem has been proved fo r the case k = 2,

we know that .i5* is a D-set in @. Therefore

I $')*Q i n ($')*)

I (IJ Q I ,,-(0 ) '

I T.\J*QI n (.1)')

Evidently T (II ~ (IJ, and so I $')*Q I ~ I T $')*Q I fur all Q EGm •

Since ,!Q* is a sub-group of $), we can spli t iQ into classes, $) '=

1..: s)*Qi' and nnw (2.1 ) follows immEdiately . The order of the group S) is kl(nl)k, and therefore

(2.2) I $') I ~ m -' ,, ($') ) ~ m-' k! (n!)k ~ n - ' (k - I) ! (n !)' .

Let sr be t he set of all permutations K with the property that the n objects of each system are transformed into objects of n different systems. In other words, ]( is such that, if a and b belong to the same system, then Ka and Kb belong to different systems. Clearly Sf is emp ty if k < n.

It is not difficult to show tha t stis a D-set. For, if H is an arbitrary p ermutation of S) , we have Sf = Sf H. It follows that Sl' is the

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sum of a number of left-classes mod ~: ~ = r K, :po Each compo­nent K, .\1 is a D-set, by theorem 1. Hence ft is aD-set.

It is easily seen that in the special case k = n t he number of ~lemcnts in st is (nl)2n, and so we have

(2.3) I Sl' 1 ~ (n!) '"n-' (k ~ n).

§ 3. T -Graphs.

In §§ 3, 4, 5, 6 we shall be mainly concerned with a special type of finite oriented linear graphs, called T-graphs 1). These have the property that, at each vertex P, the number at of oriented edges pointing to Pi equals the number of edges pointing away from Pi' For simplicity we assume a i > 0 for all i. If this number happens to be the same for all vertices (at = (J for all i) then we shall call the graph a 1'(0 ) 2).

'Vc do not exclude the possibility that, in a T-graph, several different edges point froin Pi to PI' and we neither exclude edges pointing from P; to Pi itself (closed loops).

Therefore, a T-graph can be interpreted as a pair of mappings of a finite set of edges {e l , ... , e .. J onto a finite set of vertices {PI' "', PN } such that each vertex is the image of the same number of edges in both mappings. The first mapping maps every edge onto the point \vhere it starts from, and the second one onto the point where it terminates. \Ve shall call these vertices the tail

and the head of the edge, respectively. If the head of c; coincides with the tail of el , then ci and c; will be called consecutive (which does not imply that ej and ei are consecutive).

By a complete circuit (a circnit for short ) is meant any cyclic arrangement of the set of edges in such a manner that the head of each edge coincides with the tail of the next one in the circuit. Or, in other words, such that consecutive edges in the circuit are consecutive in the graph.

Naturally, two circuits are considered as identical whenever the first one is a cyclic permutation of the second. It has to be under­stood that the order of the edges counts, and not only the order of the heads. So, fo r instance, if m = 3, N = 1, then PI is the head as well as the tail of all edges. There are two different circuits , viz. (c 1 , e2 , ea) and (e t , e3 , ez).

I) Tuttc [5 J calls them simple oriented networ/ls. 2) In the paper [IJ t he name "T-net" denoted the same thing as T (2)

does in OUf present notation.

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The number of circuits of a graph T will be denoted by I Til). A permutation P of the set of edges e1 , •.• , em will be called

conservative (with respect to T), whenever Pei = ej always implies that the head of e. coincides with the tail of ej • We choose one special conservative permutation A 0' arbitrary, but fixed in the sequel. The set of all conservative permutations of T can be repre­sented as @ A 0' where @ is the group of all permutations which leave the tails of all edges invariant.

Evidently, any circuit determines a cyclic conservative per­mutation, and vice versa.

Therefore,

I T I = I @A o '·

This simple relation between the number of circuits in a graph and the number of cyclic permutations in a set explains why we choose the same notation I I for both.

Consider a vertex Pi where a i edges start and a i edges terminate. By the local symmetric group @. we shall denote the group of all

,ail permutations which permute the at edges whose tail is Pi' but which leave invariant all edges whose tail is not Pi'

Clearly @ is the direct product of &1' ... , &N'

§ 4. Traffic regulations.

We shall also consider circuits described under certain restrictiw ,conditions, called traffic regulations.

Let mv ... , mn be sub-sets of @v ... , O:I n , respectively, and . construct the set

(4.1)

,defined in the same way as the direct product

(4.2) @ = @1 X @2 X ... X @N'

~ow a circuit described under the traffic regulation m is defined as a circuit corresponding to a permutation BAo, where BE m, and A 0 is the fixed permutation chosen in § 3.

Denoting the number of circuits described under the traffic regulation m by I T I )8, we have

(4.3) I T I = IT! 6.l , I T I \B = 1m Ao! .

1) We have ITI > 0 if and only if T is connected (see [2J). For non­·connected graphs our theorems are trivial. Nevertheless, all our proob ·are valid for that case also.

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The traffic regulation (4.1) will be called reg7tlar if, for each i, ~i is a D-set in @i 1).

The are m 2. 2) If 58 is regular, then we have 1 1

(4.4; n (58) I T I \8 = n (@) I T I (!j,

where n (58) and n (@) denote the number of elements of 58 and @, respectively.

Proof. Since @i itself satisfies the condition imposed on 58 i ,

it is sufficient to show that the value of the left-hand-side of (4.4) does not change if some 58 i is replaced by the corresponding @i'

If this has been proved, we can replace all 58 i 's by @i 's one after the other, and (4.4) follows.

To this end we consider 58, defined by (4.1) and 58*, defined by

(4.5) 58* = @1 X 582 X 58 3 X ... X 581\" and we have to show that

(4.6

for the latter ratio equals n (58) : n (58*). Referring to (4.3) we write

(4.7) IT \\8 = L: 1581 B 2 ••• BNAo I, / T I \8* = l: I @1 B 2 ... B N A 0 I,

where, in both sums, B 2 , • •• , BN run independently through the elements of 582 , ••• , 58N , respectively. If we put B2 .. . B N Ao = Q, then we have, by lemma 2,

/58 1 Q I : I @1 Q I = n (581) : n (@1)'

c'I.pplying this to each pair of corresponding terms of the sums in (4.7), We obtain (4.6).

§ 5. Special traffic regulations.

Let T be a T-graph with N vertices and m edges. Again, the numbers of edges pointing towards PI> ... , P N are denoted by aI' ... , aN' respectively, and so m = a l + ... + aN'

Let A be a positive integer. Then by TA 3) we denote the graph which arises from T if We replace any edge PiP; of T by A edges PiP;, with the same orientation. Hence TA has N vertices and Am edges. The edges of TA arising from one and the same edge of T are said to form a bundle.

') As in § 2, the bar indicates that the permutations are considered as permutations of the a i edges whose tail is Pi, whereas the other edges are disregarded.

2) This theorem was used implicitly in [1]. 3) The notations Ta and T(a) (for the latter see § 3) must not be confused.

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In T>' we shall consider several possible traffic regulations. We first choose a fixed conservative permutation A 0 which transforms bundles into bundles.

A traffic regulation will be obtained by choosing, at each \'ertex Pi' a set 58 i of permutations of the ).ai edges starting from that vertex. We shall consider three possibilities, all regular in the sense of § 4.

1°. 58 i = @i' where @i is the local symmetric group (of order (Aai )!)·

2°. 58 i = ~i' Here ~i is the sub-set of @i which transforms bundles into bundles. In other words, as to the edges whose tail is Pi it acts like the group ~ of theorem I, where the systems are given by the bundles. Thus n = A, k = ai •

3°. 58 i = Sf i . Here Sf i is the sub-set of ~i consisting of the permu­tations which transform the edges of each outgoing bundle at Pi into sets of edges belonging to A different bundles (see the end of § 2).

We have, by theorem 2,

(5.1) 1 T>'I@

iY

II (Aai )! i= 1

N

II ail (A!)CJi i= 1

N

II T (A, ai ) i=l

where T (A, ai ) is the number of clements of Sf i .

As stated in § 4, we have 1 T>' 1 Ql ~ 1 T>' 1 .

The number 1 T>' 1 ~ can be connected with the number of circuits in T itself. To this end we consider a circuit of T>' described according to the traffic regulation ~. At any stage, the bundle to which an edge belongs only depends on the bundle containing the preceding edge. Therefore, the sequence of bundles described by the circuit is periodic mod m, and any bundle is used exactly A times. It follows that each circuit under consideration defines a circuit of T. Conversely, it is easily seen that each circuit of T arises fmm A -1 (A !)m different circuits of T>' in this manner. Hence

(5.2) I T>'I ~ = 1.-1 (A!)m . I T I, and so we obtain from (5.1)

The 0 re m 3. 1 T>' 1 = A-1 . I T 1 . i§l (A:ii(!.

We shall now make the restriction that T is a T(CJ) , and that A = a, that is to say

a1 = ... = aN = a = A, m=Na.

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Then we have (see (2.3))

cp ()., O'i) = (0'!)2<7

and now (S.l) and (S.2) lead to

(S.3) 1 Ta I~ = 1 T 1.0'-1 (O'!)m . (O'~O'~l~;ar = 1 T 1.0'-1 (O'!) N(2<7-1).

If T is a T(a), with N vertices and m = O'N edges, then by T* we denote the graph defined as follows. T* has m vertices Ev ... , Em. Two vertices E i , E; are connected in T* by an edge from Ei to E j

if and only if ei , e; are consecutive in T. This process was considered in [1] for the case 0' = 2 only.

The 0 rem 4. 1 T* , = 0'-1 (0' !)N(a-l) . 1 T I.

Proof. By "O'-cycle in T" is meant a circular array containing each edge of T exactly 0' times, such that two edges are consecutive in the array if and only if they are consecutive in T. A O'-cycle will be called restricted if it is such that any pair of consecutive edges of T occurs just once as a pair of consecutive elements in the array. It will be clear that any restricted O'-cycle in T defines uniquely a circuit in T*, and vice versa.

The restricted O'-cycles in T are closely related to the circuits in Ta described under the traffic regulation ~. Actually, if we identify the edges of each bundle in Ta a ~-circuit in Ta becomes a restricted O'-cycle, owing to the definition of ~. Conversely, any restricted O'-cycle gives rise to a large number of ~-circuits. Any bundle occurs 0' times in the cycle, and each time an arbitrary edge of the bundle can be chosen.

So we see that (O'!)m different ~-circuits arise from one restricted O'-cycle. Now the theorem follows from (S.3), since m = NO'.

In a T-graph which is not necessarily a T(a) we can still consider (unrestricted) O'-cycles 1). The number of different O'-cycles can be determined from theorem 3. A difficulty lies in the fact that a O'-cycle may be periodical with a period md, where d is a proper divisor of 0', which could not happen with a restricted O'-cycle.

If c (e) denotes the number of those e-cycles in T whose period is exactly me, then we have obviously

d I T P 1 = E- c (d) (e!)m.

dip e

1) And, if T is a T(a), we can consider (unrestricted) e-cycles, for arbitrary values of e.

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Hence we obtain from Mobius' inversion formula,

c (e) = E ~ fl ([d) . (d!)-m·1 Td I, dip e

and so the number of unrestricted e-cycles equals

E c (d) = !l/ p (!l) (d !)-m d . I Td I ' dip e e d

where p is Euler's indicator. I Td I can be evaluated by theorem 3. Especially, IS T is a T(u) , then the number of unrestricted e­

cycles is

1 ( e ) ( (ad)!)N . e ~ p d (d!)u a! . 11 I .

§ 6. Trees in T-graphs.

Let T be a T-graph with N vertices and m edges. The number of edges whose tail is Pi is again denoted by a i . Choose an arbitrary vertex; for convenience of notations we take it to be Pl' We shall define the notion: (oriented) tree with root P.

A tree with root PI is a sub-set A of the set of edges of T, with the following properties. a. Any vertex #- PI is the tail of just one element of A. b. No element of A has its tail in Pl' c. Any vertex can be connected with PI by a set of consecutive

edges, all belonging to A. It is easily seen that c can be replaced by c*. A contains no closed oriented cycles.

There is a striking relation between trees and circuits. Choose a fixed edge el whose tail is P 1> and consider an arbitrary circuit of T. We traverse it, starting with el . Running through the circuit, each vertex Pi will be visited ai times. The edge by which we leave Pi after having visited it for the acth time will be called the last exit of Pi'

The 0 rem 5a. The set A consisting at the last exits at P2' ... , PN is a tree with root Pl'

Proof. The properties a and b are trivial. We shall verify c*. We can number the edges of T according to the order in the

circuit, with indices 1, ... , m; el gets the index 1. If ei and ej both belong to A, and if ei and ej are consecutive in

T, then we have i < j. For, ei+l has the same tail as el , and j is

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the maximal value of the indices of all the edges with this tail. Consequently, A does not contain any closed cycle; the indices

in such a cycle would increase indefinitely. The 0 rem 5b. If a tree A with root PI is given, and if e1

is given, then there are exactly

(6.1) N II (ai-I)! i=l

circuits of T whose set of last exits coincides with A. Proof. At any vertex we number the outgoing edges 1).

with the following restrictions: At PI the edge e1 gets the number 1; at Pi (i > 1) the edge belonging to A gets the highest possible number, that is ai . The number of ways in which this can be arranged is expressed by (6.1). It remains to be shown that, for each num­bering of this type, there exists a circuit (and not more than one circuit) corresponding with this numbering.

First thing it will be clear that we have no choice at all if we try to traverse a circuit according to this numbering. Starting with eI> we arrive at a vertex P2 , say. It is prescribed which out­going edge we have to take first, etc. If We meet a vertex for a second time, we are forced to leave it by the edge bearing the number 2, and so on. The process has to stop somewhere, the graph being finite. The only reason why it should stop is, that We arrive at a vertex where all outgoing edges have already been taken before. This must be PI> for all other vertices have been entered at least as often as they have been left.

We can show that at this moment all edges of T have been used, each exactly once of course, which means that a circuit has been described. Assume that a certain edge is vacant, that means that it has not yet been used. Considering its head, there is a vacant entry and hence there is a vacant exit. Especially, the exit belonging to A has to be vacant, since it has the highest number. This vacant edge of A leads into another vacant edge of A, and so on. Bye, we eventually arrive at PI> and we find that there is a vacant outgoing edge. This contradicts the fact that the process stopped.

From Theorem 5a and 5b we immediately obtain. The 0 rem 6. The number 0/ trees in T with a given root is

N I T I . { II (a, - I)! }-1,

i=1

1) This way of numbering is different from the one considered in t he proof of theorem Sa.

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which does not depend on the vertex chosen as the root. As before, j T I denotes the number of circuits of T.

Theorem 6 furnishes a new proof of theorem 3. For, there is a simple relation between the number of trees in T and in TA. Any tree in TA gives rise to a tree in T, by the mapping TA --+ T which maps entire bundles of TA into the corresponding edges of T. Conversely, in any bundle of TA an edge can be chosen in A ways. Any tree in T contains N-I edges, and so we have

(6.2) t (TA) = t (T) . AN-I,

where t (T) and t (TA) denote the number of trees with a given root, in T and TA, respectively. By theorem 6 we have

N

(6.3) I TA I = t (TA) . II (AG; - I) !, i=l

N (6.4) I T I = t (T) . II (G; - I) ! .

i= 1

Theorem 3 follows from (6.2), (6.3) and (6.4).

§ 7. Trees in arbitrary oriented graphs.

\Ve consider an oriented graph G, with N vertices PI' ... , PN. We no longer require that it is a T-graph, that is to say, the number of edges starting from Pi need not be the same as the number of edges pointing towards Pi'

Again, we can consider (oriented) trees, with a given root. Tutte [5J showed, that the number of trees in T with a given root can be interpreted as the value of a certain determinant. Since his result is in several ways connected with the results of the present paper, we give a full account of his theorem, with a new proof.

Let (aij) (i, j = I, ... , N) be the following matrix. If i =F- j, then a;; = - bii , where bi ; denotes the number of oriented edges from P; to Pi (Pi is the tail and Pi is the head of these edges) Further

N a ii is such that I ail = O.

1=1

The 0 rem 7 (Tutte). The number of trees with the given root Pi equals the minor of aii in the matrix (a i ;).

Proof. For simplicity of notation we take i = 1.

We first consider a special graph, where each vertex =F- PI is the tail of just one edge, and where no edge leaves Pl' This graph is

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either a tree or it is not; the possibility of constructing more than one tree in this graph does not exist. We shall show that the minor of all is 1 or ° according to whether the graph is or is not a tree.

First assume that the graph is a tree. We shall apply induction with respect to N; for N = 2 the result is trivial. Take N > 2. There is at least one vertex which is not the head of an edge. This is the case with P 2' say. Then the second column of the matrix reads 0, I, 0, ... , 0. Hence the value of the minor of an is not altered if the second row and the second column are both cancelled. The new matrix corresponds to the graph which results by cancelling P 2 and the edge starting from P 2• This new graph is still a tree, and the induction is completed.

Next assume that the graph is not a tree. Then it shows some­where a cycle of edges not containing Pl' For example, let the cycle consist of the edges P 2P a, P aP 4 , P 4P a. Then, in the matrix, the 2"d, 3rd, and 4th row are linearly dependent, for their sum vanishes. It follows that the minor of an equals zero. This completes the proof of the theorem for our special graph.

The general case is easily reduced to this one by repeated ap­plication of the following operation. Divide the set of edges starting from a certain edge, P2 , say, into two groups. Now construct two graphs; the first one arises from the original graph by cancelling the edges of the first group, the second one by cancelling the edges of the second group. The matrices of the graphs are such that the second row of the original matrix equals the sum of the correspon­ding rows in the new matrices; all other rows are identical in the three matrices. Therefore, the minor of an in the original matrix is the sum of the minors of all in the new matrices. On the other hand, the number of trees in the original graph is the sum of the numbers of trees in both graphs. This proves the theorem.

Theorem 6 shows that in a T-graph the number of trees does not depend on the choice of the root. T u t t e deduced the same fact from theorem 7. We repeat his argument. Assume that the graph considered in theorem 7 is a T-graph. Then we have a;i = a i - (li where (li is the number of edges from Pi to Pi' Therefore, we also find that the sum of the elements in each column of the matrix is equal to zero. It is a well-known fact that if in a square matrix the sum of the elements in each row and, in each column vanishes, then the cofactors of all elements have the same value. Especially, the minor of a;i does not depend on i.

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We again consider an arbitrary graph G, which need not be a T-graph. Let Pv ... , P n be its vertices, and let a i be the number of edges starting from Pi' and 'l: i the number of edges pointing towards Pi' Furthermore, bi; denotes the number of oriented edges from Pi to Pi' Hence ai = r b;;, '1:; = r bi;'

i i

Next we consider a permutation S of the N objects 1, 2, ... , N. Let Gs be the graph arising from G in the following manner: replace each edge Pi P; of Gs by an edge Pi PSi, where Sj is the result of S applied to the object j. Therefore, if the analogues of ai' ii' bi; for the graph Gs are denoted by ap), TP), bJS), respectively, then we have

(7.1) a/S) = ai , TS/S) = T; ,bi,s/S) = bi;'

Let ti (Gs) denote the number of oriented trees in Gs whose root is Pi' and let 6N denote the group of all N! permutations of the objects 1, ... , N. Then we have

The 0 rem 8. r ti (Gs) = (N - 1) ! II ak'

s.'5N k"*i

Proof. We may and do assume i = 1. We shall apply theorem 7. To this end, we consider the matrix

(i, j = 1, ... , N)

where (ji; is Kronecker's symbol. Its determinant det Ms is a multilinear polynomial in the variables Av ... , AN:

det Ms = Is (AI' ... , AN),

and it will be clear from theorem 7 that

(7.2)

We put (7.3)

() tl (Gs) = IT Is (Av a2,aa,·· .,aN).

1

In the first place we can show that P (AI' ... , AN) does not contain terms of degree < N - 1. For instance, consider the term with A3 A4 ... AN, which does not contain either Al or A2• Let T be the transposition of the objects 1 and 2. Then the coefficient of A.3 ... ),N

in det MTS is easily seen to be the opposite of the coefficient oj A3 . .• A.N in det Ms. If S runs through6N, then TS does the same, and so the coefficient of A3 • •• AN in: P (Av ... , AN) turns out tc be zero.

We next deal with the terms of degree N - 1, and therefore

162

217

we consider Al A2 ... Ai_l Ai +l ... AN. Its coefficient in Is equals - b;/SJ. Consequently, its coefficient in P (AI' ... , AN) is

- E b;/SJ = - E biS-1i = - (N - I)! Eb;; = - (N-l) !O"i S€@)N S€@)N ' j

Finally, the coefficient of Al ... AN in Is equals 1, and in P P'l'

... , AN) it is N! Thus we have proved that

P (At> ... , AN) = Al ... AN' (N -I)! {N - E O"i/A;}. i

From (7.2) and (7.3) we now deduce

() E tl (Gs) = ", P (At> 0"2' ... , O"N) = 0"2 '" O"N' (N -I)!

S€@)N 011.

Theorem 8 is, in some sense, a generalization of theorem For, if we apply theorem 8 to a graph which is a T(a), then we have, by theorem 6,

(7.4) E I Gs I = (N-I)! O"N-l {(O"-I) !}N = (N-I)! ! . (0"l)!1 S€@)N 0"

It is not difficult to see that (7.4) is equivalent with theorem 1 (take n = 0", k = N).

Note added in proof. By theorems 6 and 7 the number of circuits in a T-graph can be expressed as a determinant. For the special case that T is a T(2), this result was announced by W. T. TUTTE and C. A. B. SMITH (On unicursal paths in a network of degree 4, Amer. Math. Monthy 48, 233-237 (1941)).

REFERENCES

1. N. G. DE BRUI]N. A combinatorial problem. Nederl. Akad. Wetensch., Proc. 49, 758-764 (1946) = Indagationes Math. 8, 461-467 (1946).

2. 1. ]. GOOD. Normal recurring decimals, ]. London Math. Soc. 21, 167-169 (1947).

3. M. H. MARTIN. A problem in arrangements. Bull. Amer. Math. Soc. 40, 859-864 (1934).

4. D. REES. Note on a paper by 1. ]. GOOD. ]. London Math. Soc. 2t, 169-172 (1947).

5. \V. T. TUTTE. The dissection of equilateral triangles into equilateral triangles. Proc. Cambridge Phil. Soc. 44, 463-482 (1948).

Reprinted from Simon Slevin 28 (1951), 203-217

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