On the number of spanning trees of
a multi-complete/star related graph
Kuo-Liang Chung
a,∗, Wen-Ming Yan
b,1aDepartment of Information Management, Institute of Computer Science & Information Engineering, National Taiwan University of Science and Technology, No. 43, Section 4, Keelung Road, Taipei 10672, Taiwan
bDepartment of Computer Science and Information Engineering, National Taiwan University, No. 1, Section 4, Roosevelt Road, Taipei 10764, Taiwan
Received 14 June 1999; received in revised form 13 July 2000 Communicated by L.A. Hemaspaandra
Abstract
This paper derives a closed formula for the number of spanning trees of a multi-complete/star related graph G= Kn− Km(a1, a2, . . . , al; b1, b2, . . . , bm−l), where Km(a1, a2, . . . , al; b1, b2, . . . , bm−l) consists of l complete graphs and m− l star graphs such that the ith complete graph has ai+ 1 nodes; the jth star graph has bj+ 1 nodes, and further, the related m roots are connected together to form a complete graph. The proposed results extend previous results to a larger graph class. In addition, we provide a general maximization theorem for the multi-star graph.2000 Elsevier Science B.V. All rights reserved. Keywords: Counting; Maximization; Spanning trees; Combinatorial problems
1. Introduction
An undirected simple graph G consists of a set
V (G) of vertices and a set E(G) of edges. A complete
graph Knwith n vertices has one edge between each
pair of distinct vertices. The complement G of a graph
G= (V, E) on n vertices is defined to be the
n-vertex graph containing exactly the edges of Knwhich
are not in G. A multi-complete/star related (MCSR) graph,
G= Kn− Km(a1, a2, . . . , al; b1, . . . , bm−l), ∗Corresponding author. Supported by NSC89-2213-E011-061.
E-mail addresses: klchung@cs.ntust.edu.tw (K.-L. Chung), ganboon@csie.ntu.edu.tw (W.-M. Yan).
1Supported by NSC87-2119-M002-006.
is an n-vertex graph whose complement consists of
Km with l complete graphs and m− l star graphs
such that the ith complete graph has ai + 1 nodes;
the j th star graph has bj+ 1 nodes. In addition, the
m roots (m− l roots come from the m − l star graphs
and l roots come from the l complete graphs, any one node in the complete graph being selected as the root) are connected together to form a complete graph; the remaining k= n − m − l X i=1 ai+ m−l X j=1 bj
vertices are isolated points. Fig. 1 illustrates the graph
K3(3; 2, 3).
Some special graphs considered in [1,2,4,5] are cov-ered in the MCSR graph mentioned above. Specifi-cally, when setting l= 0, i.e., G = Kn− Km(b1, b2,
0020-0190/00/$ – see front matter 2000 Elsevier Science B.V. All rights reserved. PII: S 0 0 2 0 - 0 1 9 0 ( 0 0 ) 0 0 1 3 5 - 6
Fig. 1. K3(3; 2, 3).
. . . , bm), the MCSR graph is the so-called multi-star
related (MSR) graph. A closed formula for counting the number of spanning trees of an MSR graph was de-rived in [3] for limited m (in [6] for arbitrary m). How-ever, the technique [3,6] used in deriving the closed formulas for the number of spanning trees of an MSR graph cannot be used to derive the closed formula for the number of spanning trees of an MCSR graph straightforwardly.
Employing some new linear algebraic manipula-tions, this paper derives a closed formula for the num-ber of spanning trees of an MCSR graph. The pro-posed results cover the previous results in [1,2,4,5] and extend the results in [3,6] to a larger graph class. In ad-dition, we provide a general maximization theorem for the multi-star graph.
2. Complement spanning tree matrix of an MCSR graph
The complement spanning tree matrix (CSTM) [5]
C for a graph G is defined by
Cij=
(
n− ¯di, if i= j,
eij, if i6= j,
where ¯di is the degree of vertex i in G and eij is one
if (i, j ) is in E(G) and 0 otherwise.
From the result derived by Temperley [5], we have the following result.
Lemma 1 [5]. For the MCSR graph G, the number of spanning trees of G, τ (G), is equal to|C|/n2, where
|C| represents the determinant of matrix C.
Fig. 2. Labeling K13− K3(3; 2, 3). 3. Labeling the MCSR graph
In this section, a method is presented for labeling the MCSR graph. Given the MCSR graph G= Kn−
Km(a1, a2, . . . , al; b1, . . . , bm−l), we label the nodes
in V (Kn)−V (Km(a1, a2, . . . , al; b1, . . . , bm−l)) first.
Next we label the roots of the l complete graphs. Then we label the roots of the m− l star graphs. Further, we label the nodes in the ith complete graph except the root node for i= 1, 2, . . ., l. Finally, we label the nodes in the j th star graph except the root node for
j = 1, 2, . . ., m − l. Fig. 2 illustrates the labeling of
the complement of K13− K3(3; 2, 3).
The CSTM of Fig. 2 is equal to
C= nI2 D ! , where D= p1 1 1 1 1 1 1 p2 1 1 1 1 1 p3 1 1 1 1 α 1 1 1 1 α 1 1 1 1 α 1 β 1 β 1 β 1 β 1 β . Here, n= 13, m = 3, a1= 3, b1= 2, b2= 3, α = n− a1, β = n − 1, p1 = n − a1− m + 1, p2 =
n− b1− m + 1, and p3= n − b2− m + 1. We can
represent the matrix D in the following block-matrix form: D= P e1fT1 e2fT2 e3fT3 f1eT1 A1 f2eT2 B1 f3eT3 B2 , where fi= (1, 1, . . ., 1| {z } xi )T, ei= (0, . . ., 0, 1| {z } i , 0, . . . , 0| {z } xi−i )T,
where xi= aifor 16 i 6 l; xj= bjfor 16 j 6 m−l,
P = p1 1 1 1 p2 1 1 1 p3 , A1= α 1 1 1 α 1 1 1 α , B1= βI2, B2= βI3. 4. The closed formula
For counting the number of spanning trees of an MCSR graph as mentioned in Lemma 1, the next theorem gives our closed formula which covers the closed formula in [3,6] for counting the number of spanning trees of an MSR graph. The linear algebraic technique used in the proof is different from that used in [6].
Theorem 1. The number of spanning trees of an MCSR graph G= Kn − Km(a1, a2, . . . , al; b1, . . . , bm−l) is equal to τ (G)= nk−2 1+ a1 n− a1− 1 1+ a2 n− a2− 1 · · · · 1+ al n− al− 1 · (n − a1− 1)a1(n− a2− 1)a2· · · · (n − al− 1)al(n− 1)b1+···+bm−l|Q|, where |Q| = [1 + 1/(q1− 1) + 1/(q2− 1) + · · · + 1/(qm− 1)](q1− 1)(q2− 1) · · ·(qm− 1) where qi= n− ai− m + 1 − ai n− 1 for i= 1, 2, . . ., l, n− bi−l− m + 1 − bi−l n− 1 for i= l + 1, l + 2, . . ., m, and k= n − m − a1− a2− · · · − al− b1− · · · − bm−l. Proof. As mentioned in Section 3, the CSTM of G can be represented by C= nIk D ! , where D= P e1fT1 ··· ejfTl el+1fTl+1 ··· emfTm f1eT1 A1 . . . . .. fleTl Al fl+1eTl+1 B1 . . . . .. fmeTm Bm−l . Here P = p1 1 · · · 1 1 p2 . .. ... .. . . .. . .. 1 1 · · · 1 pm , Ai= αi 1 · · · 1 1 αi . .. ... .. . . .. . .. 1 1 · · · 1 αi ai×ai , Bi= βIbi, where αi = n − ai for 16 i 6 l, β = n − 1, pi =
n−ai−m+1 for 1 6 i 6 l, and pi = n−bi−l−m+1
for l < i6 m. From Lemma 1, the number of spanning trees of G is nk−2|D|.
Let E= Im c1f1eT1 .. . cmfmeTm ,
where c1, c2, . . . , cmare to be determined. Since
DE= P + c1e1fT1f1e1T+ · · · + cmemfTmfmeTm f1eT1+ c1A1f1eT1 .. . fleTl + clAlfleTl fl+1eTl+1+ cl+1B1fl+1eTl+1 .. . fmeTm+ cmBm−lfmeTm ,
we want to determine c1, c2, . . . , cm such that the
entries of DE are all zero except the first entry. Therefore, we have that
AifieTi = (n − 1)fieTi for i= 1, 2, . . ., l, Bi−jfieTi = (n − 1)fieTi for i= l + 1, l + 2, . . ., m. After setting ci= − 1 n− 1 for i= 1, 2, . . ., m, we have DE= Q 0 .. . 0 , where Q= P + c1e1fT1f e1T+ · · · + cmemfTmf eTm = q1 1 · · · 1 1 q2 . .. ... .. . . .. . .. 1 1 · · · 1 qm and qi= pi− ai n− 1 for i= 1, 2, . . ., l, pi− bi−l n− 1 for i= l + 1, l + 2, . . ., m. Further, we let F= Im c1e1fT1 Ia1 .. . . .. clelfT1 Ial clel+1fTl+1 Ib1 .. . . .. cmemfTm Ibm−l , then we have DF= Q e1fT1 · · · elfTl el+1fTl+1 · · · emfTm A1 . .. Al B1 . .. Bm−l .
Since the matrix DF is a block upper triangular matrix, consequently, it yields
det(D)= det(DF) = det(D) det(F )
= det(Q) det(A1)· · ·det(Al)
· det(B1)· · ·det(Bm−l).
It is observed that the matrix Q has diagonal elements q1, q2, . . . , qm and has ones on the
off-diagonals. Following the eliminating technique in [6], we border Q by adding a new first row and column. All the entries in the new row are ones, but the entries in the new column which are not in the new row are zeros. This row and column augmentation will preserve the same determinant as Q.
Subtract the new row from each of the other rows. The diagonal entries of Q in the old row indexed
−1’s except for the one in the first row. The first row
consists of all ones. Excepting the diagonal elements, the remaining elements of the matrix are zero. Now add to the first column 1/(qi − 1) times each of the
other columns to zero out the off-diagonal entries in this column. The entry in the first row and column becomes 1+ 1/(q1− 1) + 1/(q2− 1) + · · · + 1/(qm−
1). Thus, the matrix Q becomes upper triangular, so we have det(Q)= 1+ 1 q1− 1+ 1 q2− 1+ · · · + 1 qm− 1 · (q1− 1)(q2− 1) · · ·(qm− 1).
By the same argument, the matrix Ai has diagonal
elements αi, αi, . . . , αi and has ones on the
off-diagonals. It can be verified that det(Ai)= 1+ ai αi− 1 (αi− 1)ai = 1+ ai n− ai− 1 (n− ai− 1)ai for 16 i 6 l.
In addition, the matrix Bj has diagonal elements β, β,
. . . , β and has zeros on the off-diagonals, so
det(Bj)= βbj= (n − 1)bj for 16 j 6 m − l.
Consequently, the number of spanning trees of G,
τ (G), is equal to nk−2|D|. Here |D| is equal to
1+ a1 n− a1− 1 1+ a2 n− a2− 1 · · · · 1+ al n− al− 1 · (n − a1− 1)a1(n− a2− 1)a2· · · · (n − al− 1)al(n− 1)b1+···+bm−l|Q|, where |Q| = [1 + 1/(q1− 1) + 1/(q2− 1) + · · · + 1/(qm− 1)](q1− 1)(q2− 1) · · ·(qm− 1). Therefore, τ (G)= nk−2 1+ a1 n− a1− 1 1+ a2 n− a2− 1 · · · · 1+ al n− al− 1 · (n − a1− 1)a1(n− a2− 1)a2· · · · (n − al− 1)al(n− 1)b1+···+bm−l|Q|, where |Q| = [1 + 1/(q1− 1) + 1/(q2− 1) + · · · + 1/(qm−1)](q1−1)(q2−1) · · ·(qm−1). We complete the proof. 2
5. Maximization theorem for a multi-star graph In this section, a rather general maximization the-orem is provided for the multi-star graph Gn= Kn−
Km(b1, b2, . . . , bm). For the same graph, in [3], a
max-imization theorem was provided only for m= 2, 3, and 4, respectively.
From Theorem 1, considering the multi-star graph, we have
Corollary 1. The number of spanning trees of an MSR graph G= Kn− Km(b1, . . . , bm) is equal to τ (G)= nk−2(n− 1)b1+···+bm|Q|, where |Q| = [1 + 1/(q 1− 1) + 1/(q2− 1) + · · · + 1/(qm− 1)](q1− 1)(q2− 1) · · ·(qm− 1) where qi= n − bi− m + 1 − bi n− 1 for i= 1, 2, . . ., m and k= n − m − b1− · · · − bm. Since qi = n − bi− m + 1 − bi/(n− 1) and bi 6 Pm k=1bk6 n − m, we have qi= n − m + 1 − nbi n− 1 > n − m + 1 −n(n− m) n− 1 =(n− m + 1)n − (n − m + 1) − n(n − m) n− 1 =n− (n − m + 1) n− 1 =m− 1 n− 1 > 0.
We thus have qi> 0. For i6= j, it yields to
qi+ qj= 2(n − m + 1) − n(bi+ bj) n− 1 > 2(n − m + 1) −n(n− m) n− 1 = (n − m + 1) + (n − m + 1) −n(n− m) n− 1 > (n− m + 1).
Based on a reasonable assumption, there exists a positive bi. That is, we have n− m >
Pm
k=1bk> 0
Let fm(b1, b2, . . . , bm) = 1+ 1 p1+ 1 p2+ · · · + 1 pm (p1p2· · · pm), where pk= qk− 1 = n − m − nbk n− 1>−1;
the nonnegative integers b1, b2, . . . , bm satisfy the
constraint b1 + b2+ · · · + bm = L, where L is a
constant. From pi+ pj= qi− 1 + qj− 1 > 0 for all
i6= j, there is at most one negative qk for 16 k 6 m.
We now want to find the maximum of the function
fm(b1, b2, . . . , bm).
Suppose when (b1, b2, . . . , bm)= ( ˆb1, ˆb2, . . . , ˆbm),
we make function fm(b1, b2, . . . , bm) maximal. Let
ˆpk= n − m −
n ˆbk
n− 1 for k= 1, 2, . . ., m.
The maximization theorem for the multi-star graph is proved by the technique of contradiction. Suppose
ˆbi> ˆbj+ 1. We consider two cases. In case 1, if there
exists an l such that ˆpl< 0, then we have ˆbl> ˆbi, i.e.,
ˆbl> ˆbj+ 1. For all k 6= l, j, we thus have ˆpk> 0. In
case 2, for all k6= i, j, we have ˆpk> 0.
By the symmetric property of fm(b1, b2, . . . , bm),
for case 1, we assume l= 1 and j = 2 for convenience and for case 2, we assume i= 1 and j = 2. Hence we have ˆb1> ˆb2+ 1 and for all k > 3, we have ˆpk> 0.
The maximum of fm(b1, b2, . . . , bm) is given by
fm( ˆb1, ˆb2, . . . , ˆbm)= fm−2( ˆb3, ˆb4, . . . , ˆbm)(ˆp1ˆp2)
+ ( ˆp1+ ˆp2)(ˆp3ˆp4· · · ˆpm).
Since each ˆpk > 0 for k = 3, 4, . . ., m, we have
fm−2(ˆp3, ˆp4, . . . ,ˆpm) > 0. Now let ˆp10 = n − m − n(b1− 1) n and ˆp20 = n − m − n(b2+ 1) n ,
then we have ˆp10 + ˆp02= ˆp1+ ˆp2and
ˆp10 ˆp02=14 (ˆp01+ ˆp02)2− ( ˆp01− ˆp02)2 >14(ˆp1+ ˆp2)2− ( ˆp1− ˆp2)2 = ˆp1ˆp2.
From the above inequality, we have
fm( ˆb1− 1, ˆb2+ 1, . . ., ˆbm) = fm−2( ˆb3, ˆb4, . . . , ˆbm)(ˆp01ˆp02) + ( ˆp10+ ˆp20)(ˆp3ˆp4· · · ˆpm) > fm−2( ˆb3, ˆb4, . . . , ˆbm)(ˆp1ˆp2) + ( ˆp1+ ˆp2)(ˆp3ˆp4· · · ˆpm) = fm( ˆb1, ˆb2, . . . , ˆbm).
It is a contradiction. That is, the assumption ˆb1 >
ˆb2+ 1 is false. On the contrary, we must have ˆb16
ˆb2+ 1. Generally, considering all pair of i and j and
from Corollary 1, we have the following maximization theorem for the multi-star graph.
Theorem 2. The number of spanning trees of an MSR graph G= Kn−Km(b1, . . . , bm) is maximal when the difference between bi and bj for 16 i, j 6 m is at most one.
6. Conclusion
Graphs are often used to model network problems. For example, we often use star graphs (like MSR), complete graphs, or hybrid graphs (like MCSR) to model connection patterns in a network. Among these graphs, finding a spanning tree with some specific properties, such as minimum weight and minimum label, is a well-known technique to solve the net-work routing problem. Since the number of spanning trees in a graph could reveal the complexity of the corresponding combinatorial configuration, deriving a closed form of the number of spanning trees is in-deed a research problem. Since the CSTM form of the MCSR graph is much more complicated than that of the MSR graph, deriving the corresponding closed form of the MCSR is much harder than that of the MSR graph.
We have presented how to derive the closed form for the number of spanning trees of an MCSR graph G=
Kn − Km(a1, a2, . . . , al; b1, . . . , bm−l) for arbitrary
m. The main contribution of this paper is that the
derived closed formula extends that of the graph class discussed in the previous results [3,6] of the MSR graph to a larger MCSR graph class. Another minor contribution of this paper is that a general
maximization theorem is provided for the multi-star graph.
Acknowledgement
The authors are indebted to the anonymous referees and editor Professor Lane A. Hemaspaandra for their valuable suggestions that lead to the improved version of the paper.
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