Enumerating spanning trees of graphs with an involution

Journal of Combinatorial Theory Ser. A, 116(2009), 650-662 ENUMERATING SPANNING TREES OF GRAPHS WITH AN INVOLUTION FUJI ZHANG AND WEIGEN YAN Abstract. As the extension of the previous work by Ciucu and the present authors (J. Combin. Theory Ser. A 112(2005) 105–116), this paper considers the problem of enumeration of spanning trees of weighted graphs with an involution which allows fixed points. We show that if G is a weighted graph with an involution, then the sum of weights of spanning trees of G can be expressed in terms of the product of the sums of weights of spanning trees of two weighted graphs with a smaller size determined by the involution of G. As applications, we enumerate spanning trees of the almost-complete bipartite graph, the almost-complete graph, the Möbius ladder, and the almost-join of two copies of a graph. 1. Introduction Inspired by Ciucu’s result which expressed the number of perfect matchings of a plane bipartite graph with reflective symmetry in terms of the numbers of perfect matchings of two smaller graphs, Yan and Zhang [12, 13] extended this to the general plane graphs (not necessarily bipartite) with reflective symmetry. Ciucu, Yan, and Zhang [6] considered the problem of enumeration of spanning trees of plane graphs with reflective symmetry and obtained a similar result to that in Ciucu [5]. For many mathematicians and chemists, they are interested in not only the plane graphs but also the nonplanar graphs with reflective symmetry (see for example [3, 9, 13]). Graham et al [7] and Chen [4] considered the other problems for the graph with an antipodal automorphism and the so-called k-pairable graph, respectively. In this paper, we consider the more general case of the weighted graph with involution which allowed fixed points. Let G = (V (G), E(G)) be a connected graph with an involution f . Hence there exist three disjoint subsets of V (G), denoted by VL = {v1 , v2 , . . . , vs }, VC = 2000 Mathematics Subject Classification. Primary 05C15, 05C16. Key words and phrases. Involution, spanning tree, Laplacian matrix, Matrix-Tree Theorem, Laplacian eigenvalue. The first author was supported in part NSFC Grant #10671162. The second author was supported in part by NSFC Grant #10771086 and by Program for New Century Excellent Talents in Fujian Province University. 1 2 FUJI ZHANG AND WEIGEN YAN {v1′′ , v2′′ , . . . , vn′′ }, and VR = {v1′ , v2′ , . . . , vs′ }, such that V (G) = VL ∪ VC ∪ VR (VC can be the empty set, and the partition may not be unique) and the following properties hold: (i). Let GL and GR be two subgraphs of G induced by VL and VR , respectively. Then σ : vi 7−→ vi′ for 1 ≤ i ≤ s is an isomorphism between GL and GR . (ii). The involution f satisfies: f (vi ) = vi′ , f (vi′ ) = vi , and f (vj′′ ) = vj′′ for 1 ≤ i ≤ s, 1 ≤ j ≤ n. Suppose G is a connected edge-weighted graph (For convenience, we say a connected weighted graph) and the weight of each edge e of G is denoted by w(e) which is a real number. If the underlying graph of G has an involution f and for every e ∈ E(G) we have w(e) = w(f (e)), that is, the weighted function on the edges is constant on the orbits of f , then we say that G is a weighted graph with an involution f (or f preserves weights). Obviously, a plane graph with reflective symmetry defined in [5, 6] has an involution. Suppose that G = (V (G), E(G)) is a weighted graph with no multiple edges and no loops and with vertex set V (G) and edge set E(G). The weight w(T ) of a spanning tree T in G is defined as the product of weights of edges in T , i.e., Y w(e). w(T ) = e∈E(T ) The sum of weights of spanning trees of G is denoted by t(G). Hence X t(G) = w(T ), T where the summation is over all spanning trees of G. In this paper, we consider the problem of enumeration of spanning trees of weighted graphs with an involution and we prove that if G is a connected weighted graph with an involution f , then t(G) can be expressed in terms of t(GL ) and t(GR ), where GL and GR are two smaller graphs which are determined by f and will be defined later. Our result generalizes the one obtained in [6]. As applications, we enumerate spanning trees of some graphs. Some related results of enumeration of spanning trees see for example [1, 2, 3, 6, 8, 10, 11, 14]. 2. Main result Suppose that G = (V (G), E(G)) is a connected weighted graph with an involution f . The graph illustrated in Figure 1(a) has an involution f . Hence V (G) can be partitioned into V (G) = VL ∪ VC ∪ VR , where VL = {v1 , v2 , . . . , vs }, ENUMERATING SPANNING TREES OF GRAPHS WITH AN INVOLUTION 3 Figure 1. (a) A connected graph G with an involution f such that f (vi ) = vi′ , f (vi′ ) = vi , and f (vj′′ ) = vj′′ for 1 ≤ i ≤ 4, 1 ≤ j ≤ 3, where the left part GL (resp. right part GR ) of G is the subgraph induced by {v1 , v2 , v3 , v4 } (resp. {v1′ , v2′ , v3′ , v4′ }) and the center part GC of G is the subgraph induced by {v1′′ , v2′′ , v3′′ }. (b) The weighted graph GL obtained from G. (c) The weighted graph GR obtained from G. VC = {v1′′ , v2′′ , . . . , vn′′ }, and VR = {v1′ , v2′ , . . . , vs′ } (VC can be the empty set). Let GL and GR be two weighted subgraphs of G induced by VL and VR , respectively. Without loss of generality, we call GL and GR the left part and right part of G (see Figure 1), respectively. Similarly, call GC , which is the weighted subgraph of G induced by VC , the center part of G. We can partition the edge set of G as follows: E(G) = E(GL ) ∪ E(GR ) ∪ E(GC ) ∪ E(GL − GR ) ∪ E(GL − GC ) ∪ E(GR − GC ), where E(GL − GR ) denotes the set of edges in G between GL and GR , and E(GL − GC ) (or E(GR − GC )) is the set of edges in G between GL and GC (or between GR and GC ). Now we construct two new weighted graphs GL and GR from G as follows. Let GL be the weighted graph obtained from GL by the following four procedures (see Figure 1(b)): 1. Add a new vertex u to GL . 2. For each edge e = (vi , vj′ ) ∈ E(GL −GR ) with weight w(e), add an edge (u, vi ) in GL with weight 2w(e). 3. For each pair of edges e = (vi , vj′ ) and e′ = (vj , vi′ ) (where i 6= j) in E(GL − GR ), add an edge (vi , vj ) in GL with weight −w(e) (since G has an involution, if e = (vi , vj′ ) is an edge with weight w(e) in G, then there exists an edge e′ = (vj , vi′ ) with weight w(e′ ) = w(e) in G). 4 FUJI ZHANG AND WEIGEN YAN 4. For each edge e = (vi , vj′′ ) ∈ E(GL − GC ), add an edge (u, vi ) in GL with weight w(e). Obviously, GL may contain multiple edges. For the graph G in Figure 1(a), there exist five edges (v1 , v2′ ), (v2 , v1′ ), (v3 , v3′ ), (v3 , v4′ ), and (v4 , v3′ ) in E(GL − GR ). Hence, by the above Procedure 2, in GL there exist one edge with weight 2 between u and v1 , one edge with weight 2 between u and v2 , two edges with weight 2 between u and v3 , and one edge with weight 2 between u and v4 (see Figure 1(b)). Note that there exist two pairs of edges in E(GL − GR ), which have the form (vi , vj′ ) and (vj , vi ) (i 6= j), i.e., (v1 , v2′ ) and (v2 , v1′ ), and (v3 , v4′ ) and (v4 , v3′ ) (see Figure 1(a)). Hence, by the above Procedure 3, in GL there exist two edges (v1 , v2 ) and (v3 , v4 ) with weight −1 (see Figure 1(b)). Hence, for the graph illustrated in Figure 1(a), the corresponding graph GL is illustrated in Figure 1(b). Let GR be the weighted graph obtained from the weighted subgraph of G induced by VC ∪ VR by the following two procedures (see Figure 1(c)): 1′ . Reduce the weight of each edge in GC by half. 2′ . For each pair of edges e = (vi , vj′ ) and e′ = (vj , vi′ ) (where i 6= j) in E(GL − GR ), add an edge (vi′ , vj′ ) with weight w(e) (since G has an involution, if e = (vi , vj′ ) is an edge with weight w(e) in G, then there exists an edge e′ = (vj , vi′ ) with weight w(e′ ) = w(e) in G). For the graph G in Figure 1(a), the corresponding graph GR is illustrated in Figure 1(c). Now we can state our main result as follows. Theorem 2.1. Suppose that G = (V (G), E(G)) is a weighted graph with an involution f and GC , GL , and GR are defined as above. Then the sum of weights of spanning trees of G is given by t(G) = 2n−1 t(GL )t(GR ), where n is the number of vertices of GC . Using Lemma 6 in [6] it is not difficult to see that Theorem 4 in [6] is a special case of Theorem 2.1 above. 3. Proof In order to prove Theorem 2.1, we need to introduce some notation and terminology as follows. Suppose that G = (V (G), E(G)) is a weighted graph with no multiple edges and no loops, with vertex set V (G) = {v1 , v2 , · · · , vν } and edge set E(G) = {e1 , e2 , · · · , em }, if not otherwise specified. Define di (G) to be the sum ENUMERATING SPANNING TREES OF GRAPHS WITH AN INVOLUTION 5 of weights of edges incident with vertex vi . Let the diagonal matrix of vertex degrees of G be D(G) = diag(d1 (G), d2 (G), . . . , dν (G)). Denote the adjacency matrix of G by A(G) = (aij )ν×ν and the Laplacian matrix of G by L(G) = D(G) − A(G), where aij equals the weight of edge (vi , vj ) if (vi , vj ) is an edge of G and aij equals zero otherwise (if G is a weighted graph with multiple edges, then aij equals the sum of weights of edges between vertices vi and vj if there exist edges between vi and vj and aij equals zero otherwise). A well-known formula for t(G), the sum of weights of spanning trees of G, is “the Matrix-Tree Theorem” (see e.g. Biggs [1] or Bondy and Murty [2]), which expresses t(G) as a determinant. Theorem 3.1 (the Matrix-Tree Theorem). Suppose that G is a weighted graph with ν vertices and L(G) is the Laplacian matrix of G. Then the sum of weights of spanning trees of G is t(G) = det(L(G)i ), where L(G)i is the submatrix of L(G) obtained by deleting the i-th row and the i-th column from L(G) for any 1 ≤ i ≤ ν. − → Let G = (V, A) be a weighted digraph with no multiple arcs and no loops, with vertex set V = {v1 , v2 , · · · , vν }. Denote the sum of weights of arcs with − → − → initial vertex vi by di ( G ). Let the diagonal matrix of vertex degrees of G be − → − → − → − → − → D( G ) = diag(d1 ( G ), d2 ( G ), . . . , dν ( G )), and let A( G ) = (bij )ν×ν be the adjacency − → − → − → − → − → matrix of G and L( G) = D( G ) − A( G ) the Laplacian matrix of G , where bij − → − → equals the weight of (vi , vj ) if (vi , vj ) is an arc of G and zero otherwise (if G is a weighted digraph with multiple arcs, then bij equals the sum of weights of arcs with initial vertex vi and terminal vertex vj if there exist such arcs and bij equals zero − → otherwise). A directed spanning tree of G with root vi is a rooted spanning tree, − → endowed with the induced orientation of the diagraph G , such that its branches are − → oriented toward the root. Denote by t( G i ) the sum of weights of directed spanning Pν − → − → trees of G with root vi and by t( G ) = i=1 t( G i ) the sum of weights of directed − → spanning trees of G . A corresponding result to Theorem 3.1 is the following − → Theorem 3.2 (Tutte, 1948). Suppose that G is a weighted digraph with ν vertices − → − → and L( G ) is the Laplacian matrix of G . Then − → − → t( G i ) = det(L( G )i ), − → − → where L( G )i is the submatrix of L( G ) obtained by deleting the i-th row and the − → i-th column from L( G ) for any 1 ≤ i ≤ ν. The following two lemmas are well known (see for example [1]). 6 FUJI ZHANG AND WEIGEN YAN Lemma 3.3. Let 0 < µ1 ≤ µ2 ≤ . . . ≤ µν−1 be the Laplacian spectrum of a connected weighted graph G with ν vertices. Then the sum of weights of spanning trees of G is t(G) = µ1 µ2 . . . µν−1 . ν Lemma 3.4. Let 0 < µ1 ≤ µ2 ≤ . . . ≤ µν−1 be the Laplacian spectrum of a − → connected weighted digraph G with ν vertices. Then the sum of weights of directed − → spanning trees of G is − → t( G ) = µ1 µ2 . . . µν−1 . Figure 2. The digraph G′R obtained from G in Figure 1(a), where each of the edges without orientation in the figure is replaced with two oppositely oriented arcs. Suppose that G = (V (G), E(G)) is a weighted graph with an involution f and GL , GR , GC , GL , and GR are defined as above. Let G∗R be the weighted graph obtained from the weighted subgraph of G induced by VC ∪VR by procedure 2′ only (G∗R and GR have the same underlying graph and wG∗ (e) = 2wGR (e) if e ∈ E(GC ) R and wG∗ (e) = wGR (e) otherwise, where wG (e) denotes the weight of edge e in G). R Let G′R be the weighted digraph obtained from G∗R by replacing each edge e = (u, v) with two arcs (u, v) and (v, u) with weights equal to w(e). Let G′R be the weighted digraph obtained from G′R by adding an arc (vi′′ , vj′ ) with weight w(e) for each edge e = (vi′′ , vj′ ) ∈ E(GC − GR ). For the graph G in Figure 1(a), the corresponding digraph G′R is illustrated in Figure 2, where each edge without orientation in the figure is replaced with two oppositely oriented arcs. The following lemma will play a key role in the proof of our main result. Lemma 3.5. Suppose that G = (V (G), E(G)) is a weighted graph with an involution f and GL , GR , GC , GL , and G′R are defined as above. Then the weighted ENUMERATING SPANNING TREES OF GRAPHS WITH AN INVOLUTION 7 enumeration of all spanning trees of G is given by t(G) = 1 t(GL )t(G′R ), |V (G)| where |V (G)| is the number of vertices of G and t(G′R ) is the sum of weights of directed spanning trees of G′R . ′ Proof Given GL , GR , GC , GL and GR , |V (GL )| = |V (GR )| = s, and |V (GC )| = n. Hence 2s + n = |V (G)| =: ν. Now G has an involution f and G satisfies the properties (i) and (ii). By a suitable labelling of vertices of G, the adjacency matrix of G has the following form:  M B  T A(G) =   B S FT B F   BT  , M where M is the adjacency matrix of GL or GR , S is the adjacency matrix of GC , B denotes the incident relation between GL and GC , and F denotes the incident relation between GL and GR . Since G satisfies the property (ii) and f is a weightpreserving involution, we have F = F T . Hence  M B  T A(G) =   B S F   Let D(G) =   B F   BT  . M  DL   be the corresponding diagonal matrix of vertex  DC DR degrees of G, where DL (= DR ) and DC are two diagonal matrices of order s and n, respectively. Hence the Laplacian matrix of G   L(G) =   DL − M −B T −F −B DC − S and the characteristic polynomial of L(G) −B −F −B T DL − M   .  8 FUJI ZHANG AND WEIGEN YAN   det(xIν − L(G))= det   = = =   det     det   xIs − DL + M BT F xIs − DL + F + M 2B T xIs − DL + F + M xIs − DL + F + M 2B T 0 B F xIn − DC + S BT B B xIn − DC + S B B xIn − DC + S 0 det(xIs − DL + M − F ) det " xIs − DL + M  F   BT  xIs − DL + M  F   BT  xIs − DL + M − F xIs − DL + F + M 2B T B xIn − DC + S     # . Set φ(x) ψ(x) = = det(xIs − DL + M − F ), " xIs − DL + F + M det 2B T Note that B xIn − DC + S # . (3.1) d d det(xIν − L(G)) = (φ(x)ψ(x)) = φ′ (x)ψ(x) + φ(x)ψ ′ (x). dx dx Hence, by Lemma 3.3, νt(G) = µ1 µ2 · · · µν−1 d [det(xIν − L(G))]|x=0 = (−1)ν−1 dx (3.2) = (−1)ν−1 [φ′ (0)ψ(0) + φ(0)ψ ′ (0)], where µ1 , µ2 , . . ., and µν−1 are the nonzero Laplacian eigenvalues of G. From (3.1) and (3.2) it suffices to prove the following Claims 1-3: Claim 1. ψ(0) = 0. Claim 2. φ(0) = (−1)s t(GL ). Claim 3. ψ ′ (0) = (−1)s+n−1 t(G′R ). We need to prove the following claims: Claim 4. The Laplacian matrix of the digraph G′R equals " DL − F − M −2B T −B DC − S Claim 5. DL − M + F is the submatrix of the Laplacian matrix L(GL ) of GL obtained by deleting the row and column corresponding to vertex u. " # DL − F − M −B First we prove Claim 4. Let X = (xij )1≤i,j≤s+n = . −2B T DC − S Set δij = 1 if i = j and δij = 0 otherwise. Note that xij = δij dG (vi′ ) − fij − mij if 1 ≤ i, j ≤ s and xi+s,j+s = δij dG (vj′′ ) − sij if 1 ≤ i, j ≤ n, and xi,j+s = −bij # . ENUMERATING SPANNING TREES OF GRAPHS WITH AN INVOLUTION 9 for 1 ≤ i ≤ s, 1 ≤ j ≤ n and xi+s,j = −2bji for 1 ≤ i ≤ n, 1 ≤ j ≤ s, where F = (fij )1≤i,j≤s , M = (mij )1≤i,j≤s , S = (sij )1≤i,j≤n , and B = (bij )1≤i≤s,1≤j≤n . For 1 ≤ i ≤ s, xii = dG (vi′ )−fii (since M is the adjacency matrix of GR , we have mii = 0) and fii equals the weight of edge (vi , vi′ ) in G. Hence, by the definition of G′R , xii is the sum of weights of arcs with initial vertex vi′ in G′R . For 1 ≤ i ≤ n, xi+s,i+s = dG (vi′′ ) − sii = dG (vi′′ ). By the definition of G′R , dG (vi′′ ) is exactly the sum of weights of arcs with initial vertex vi′′ in G′R . So we have proved the following: Claim 4.1. For 1 ≤ i ≤ s + n, diag(x11 , x22 , . . . , xs+n,s+n ) is the diagonal matrix of vertex out-degrees of G′R . For 1 ≤ i, j ≤ s, i 6= j, −xij = fij + mij . Since mij is the weight of edge (vi′ , vj′ ) in G and fij is the weight of edge (vi′ , vj ) (or (vi , vj′ )) in G, by the definition of G′R , fij + mij equals the sum of weights of arcs with initial vertex vi′ and terminal vertex vj′ . For 1 ≤ i ≤ s, 1 ≤ j ≤ n, −xi,j+s = bij , −xi+s,j = 2bji , and −xi+s,j+s = sij (i 6= j). Note that bij equals the weight of edge (vi , vj′′ ) (or (vi′ , vj′′ )) in G. By the definition of G′R , −xi,j+s = bij equals the weight of arc (vi′ , vj′′ ) in G′R and −xi+s,j = 2bji equals the sum of weights of arcs with initial vertex vi′′ and terminal vertex vj′ in G′R . Similarly, −xi+s,j+s = sij (i 6= j) is the weight of edge (vi′′ , vj′′ ) in G′R which also equals the weight of arc (vi′′ , vj′′ ) in G. Hence we have proved the following: Claim 4.2. diag(x11 , x22 , . . . , xs+n,s+n ) − (xij )1≤i,j≤s+n is the adjacency matrix of G′R . Claim 4 is immediate from Claims 4.1 and 4.2. Now we prove Claim 5. Let Y = (yij )1≤i,j≤s = DL − M + F . We need to prove that yii for 1 ≤ i ≤ s is the sum of weights of edges incident with vertex vi in GL (i.e., dGL (vi ) = yii ) and −yij for 1 ≤ i 6= j ≤ s is the sum of weights of edges joining vertices vi to vj in GL . Note that yii = dG (vi ) − mii + fii = dG (vi ) + fii . Let NG (vi ) be the set of vertices of G incident with vertex vi . Suppose NG (vi ) = {vix |1 ≤ x ≤ l} ∪ {vj′ y |1 ≤ y ≤ p} ∪ {vk′′z |1 ≤ z ≤ q}. Then dG (vi ) = l X x=1 w(ex ) + p X y=1 w(e′y ) + q X w(e′′z ), z=1 where ex = (vi , vix ), e′y = (vi , vj′ y ), and e′′z = (vi , vk′′z ). By the definition of GL , NGL (vi ) = {vix |1 ≤ x ≤ l}∪{u}∪{vjy |1 ≤ y ≤ p, jy 6= i} and there exist p+q multi- ple edges between vertices u and vi with weights 2w(e′1 ), 2w(e′2 ), . . . , 2w(e′p ), w(e′′1 ), 10 FUJI ZHANG AND WEIGEN YAN w(e′′2 ), . . . , w(e′′q ) in GL , respectively. Moreover, the weight of adding edge (vi , vjy ) in GL equals −w(e′y ) for i 6= jy , 1 ≤ y ≤ p. Hence we have dGL (vi ) = l P w(ex ) + x=1 p P y=1 2w(e′y ) + = dG (vi ) + w((vi , vi′ )) q P z=1 w(e′′z ) − P 1≤y≤p,jy 6=i w(e′y ) (3.3) = dG (vi ) + fii . For 1 ≤ i 6= j ≤ s, −yij = mij − fij . Note that if fij 6= 0, then there exists an edge (vi , vj′ ) in G with weight fij . By the definition of GL , we added a new edge (vi , vj ) with weight −fij . Hence the sum of weights of edges joining vi and vj in GL equals mij − fij = −yij . So we have proved Claim 5. Claim 1 follows from (3.1) and Claim 4, Claim 2 follows from Theorem 3.2 and Claim 5, and Claim 3 is immediate from (3.1), Lemma 3.2, and Claim 4. Hence we have completed the proof of the lemma.  Proof of Theorem 2.1. We use the notation in the proof of Lemma 3.5. We have proved the following: t(G) = where 1 t(GL )t(G′R ), ψ ′ (0) = (−1)s+n−1 t(G′R ), |V (G)| ψ(x) = det " xIs − DL + F + M " xIs − DL + F + M 2B T Note that n ψ(x) = 2 det BT B xIn − DC + S # B x 2 In − 21 DC + 21 S (3.4) . # . Hence we have ψ ′ (0) = (−1)s+n−1 2n s X det(Qi ) + (−1)s+n−1 2n−1 det(Qj ), (3.5) j=s+1 i=1 where Qi is the matrix obtained from s+n X " DL − F − M −B # by deleting 1 1 −B T 2 DC − 2 S the i-th row and the i-th column. We now prove the following: # " DL − F − M −B . Claim 6. The Laplacian matrix of GR equals 1 1 −B T 2 DC − 2 S Note that L(G′R ) is obtained from L(G′R ) by dividing the lower left block by 2 (direct from the definition) and L(G′R ) = L(G∗R ) (also from the definition, since it is just the replacement of undirected edges by two arcs in opposite direction). And finally, to obtain L(GR ), one should apply 1′ , what corresponds to dividing the lower right block by 2. Hence Claim 6 holds. ENUMERATING SPANNING TREES OF GRAPHS WITH AN INVOLUTION 11 By Theorem 3.2 and Claim 6, for 1 ≤ i ≤ s + n, we have det(Qi ) = t(GR ). Hence, by (3.5), ψ ′ (0) = (−1)s+n−1 2n−1 [2s × t(GR ) + n × t(GR )] = (−1)s+n−1 |V (G)|2n−1 t(GR ). (3.6) The theorem follows from (3.4) and (3.6).  4. Applications In this section, as applications of Theorem 2.1, we enumerate spanning trees of the almost-complete bipartite graph, the almost-complete graph, the Möbius ladder, and the almost-join of two copies of a graph. 4.1. The almost-complete bipartite graph. Let G be a graph constructed by removing p disjoint edges from the complete bipartite graph Kn,n (p ≤ n). Without loss of generality, we can assume that V (G) = {v1 , v2 , . . . , vn } ∪ {v1′ , v2′ , . . . , vn′ } and E(G) = {(vi , vj′ )|1 ≤ i, j ≤ n} \ {(vi , vi′ )|1 ≤ i ≤ p}. Let f : V (G) −→ V (G) satisfy f (vi ) = vi′ and f (vi′ ) = vi for 1 ≤ i ≤ n. Obviously, f is an involution of G satisfying VC = ∅. By the definitions of GR and GL , GR is the complete graph Kn and GL is a weighted complete graph with vertex set {v1 , v2 , . . . , vn } ∪ {u}, where the weight of each edge (vi , vj ) (i 6= j) in GL equals −1 and the weight of edge (vi , u) equals 2(n − 1) for 1 ≤ i ≤ p and 2n for p + 1 ≤ i ≤ n. Note that, for vi ∈ V (GL ), we have dGL (vi ) = 2(n−1)−(n−1) = n−1 for 1 ≤ i ≤ p and dGL (vi ) = 2n−(n−1) = n+1 for p + 1 ≤ i ≤ n. Hence the submatrix obtained from the Laplacian matrix of GL by deleting the row and column corresponding to vertex u is the following: " # (n − 1)Ip + (Jp,p − Ip ) Jp,n−p A= , Jn−p,p (n + 1)In−p + (Jn−p,n−p − In−p ) where Ip and Ji,j denote the identity matrix of order p and the i × j matrix with each entry equal to one, respectively. It is not difficult to prove the following: det(A) = 2[(n − 2)n + p](n − 2)p−1 nn−p−1 . By the Matrix-Tree Theorem, we have t(GL ) = 2[(n − 2)n + p](n − 2)p−1 nn−p−1 . (4.1) Hence, by Theorem 2.1, we have t(G) = 21 t(GL )t(GR ) = = 1 2 × {2[(n − 2)n + p](n − 2)p−1 nn−p−1 } × nn−2 [(n − 2)n + p] (n − 2)p−1 n2n−p−3 . 12 FUJI ZHANG AND WEIGEN YAN Thus we have proved the following: Theorem 4.1. Let G be a graph constructed by removing p disjoint edges from the complete bipartite graph Kn,n (p ≤ n). Then t(G) = [(n − 2)n + p] (n − 2)p−1 n2n−p−3 . Particularly, if p = n we have t(G) = (n − 1)(n − 2)n−1 nn−2 . 4.2. The almost-complete graph. Let G be a graph constructed by removing q disjoint edges from a complete graph Kn , where n ≥ 2q. Then the number of spanning trees of G  q 2 1− t(G) = n . (4.2) n This formula can be found on page 43 in [1]. Now we use Theorem 2.1 to give a n−2 new proof of this formula. We need to distinguish two cases as follows. Case 1. n is even. Let n = 2k, k ≥ q. For convenience, let V (G) = {v1 , v2 , . . . , vk } ∪ {v1′ , v2′ , . . . , vk′ } and E(G) = {(vi , vj )|1 ≤ i 6= j ≤ k} ∪ {(vi′ , vj′ )|1 ≤ i 6= j ≤ k} ∪ {(vi , vj′ )|1 ≤ i, j ≤ k} \ {(vi , vi′ )|1 ≤ i ≤ q}. Let f : V (G) −→ V (G) satisfy f (vi ) = vi′ and f (vi′ ) = vi for 1 ≤ i ≤ k. Obviously, f is an involution of G satisfying VC = ∅. By the definition of GL , GL is isomorphic to a weighted star with vertex set V (GL ) = {v1 , v2 , . . . , vk }∪{u} and edge set E(GL ) = {(vi , u)|1 ≤ i ≤ k}, where the weight of edge (vi , u) is 2(k − 1) if 1 ≤ i ≤ q and 2k if q + 1 ≤ i ≤ k. Hence we have t(GL ) = [2(k − 1)]q (2k)k−q = 2k (k − 1)q k k−q . (4.3) Similarly, by the definition of GR , GR is a weighted complete graph with k vertices, where the weight of each edge equal two. Hence t(GR ) = k k−2 2k−1 . By Theorem 2.1, we have 1 2 q 1 t(GL )t(GR ) = 2k (k − 1)q k k−q k k−2 2k−1 = (2k)2k−2 (1 − ) 2 2 2k implying that (4.2) holds if n is even. t(G) = Case 2. n is odd. Let n = 2k + 1, k ≥ q. We assume V (G) = {v1 , v2 , . . . , vk } ∪ ′ {v1 , v2′ , . . . , vk′ } ∪ {v1′′ } and E(G) = {(vi , vj )|1 ≤ i 6= j ≤ k} ∪ {(vi′ , vj′ )|1 ≤ i 6= j ≤ k} ∪ {(v1′′ , vi ), (v1′′ , vi′ )|1 ≤ i ≤ k} ∪ {(vi , vj′ )|1 ≤ i, j ≤ k} \ {(vi , vi′ )|1 ≤ i ≤ q}. Let f : V (G) −→ V (G) satisfy f (vi ) = vi′ and f (vi′ ) = vi for 1 ≤ i ≤ k, and f (v1′′ ) = v1′′ . Obviously, f is an involution of G satisfying VC = {v1′′ }. By the definition of GL , GL is isomorphic to a weighted star with vertex set V (GL ) = {v1 , v2 , . . . , vk } ∪ {u} ENUMERATING SPANNING TREES OF GRAPHS WITH AN INVOLUTION 13 and edge set E(GL ) = {(vi , u)|1 ≤ i ≤ k}, where the weight of edge (vi , u) is 2k − 1 if 1 ≤ i ≤ q and 2k + 1 if q + 1 ≤ i ≤ k. Hence we have t(GL ) = (2k − 1)q (2k + 1)k−q . (4.4) Similarly, by the definition of GR , GR is a weighted complete graph with vertex set {v1′′ , v1′ , v2′ , . . . , vk′ }, where the weight of each edge (v1′′ , vj′ ) for 1 ≤ j ≤ k is one and the weight of each edge (vi′ , vj′ ) for 1 ≤ i 6= j ≤ k is two. It is not difficult to see that the submatrix obtained from the Laplacian matrix of GR by deleting the row and column corresponding to vertex v1′′ has the following form:     A=   2k − 1 −2 −2 .. . 2k − 1 .. . −2 −2 ··· ··· .. . ··· −2 −2 .. . 2k − 1        . k×k From the Matrix-Tree Theorem, we have t(GR ) = det(A) = (2k + 1)k−1 . Hence, by Theorem 2.1, we have t(G) = 21−1 t(GL )t(GR ) = (2k − 1)q (2k + 1)k−q (2k + 1)k−1  q 2 = (2k + 1)2k−1 1 − 2k+1
q = nn−2 1 − n2 implying that (4.2) holds if n is odd. Figure 3. (a) The Möbius ladder M (n). (b) The weighted graph M (n)L . (c) The weighted graph M (n)R . 14 FUJI ZHANG AND WEIGEN YAN 4.3. The Möbius ladder. The Möbius ladder M (n) (n ≥ 3) is a regular graph of degree 3 with 2n vertices (see Figure 3(a)), with vertex set V (M (n)) = {v1 , v2 , . . . , vn }∪ ′ {v1′ , v2′ , . . . , vn′ } and E(M (n)) = {(vi , vi+1 )|1 ≤ i ≤ n − 1} ∪ {(vi′ , vi+1 )|1 ≤ i ≤ n − 1} ∪ {(v1 , vn′ ), (vn , v1′ )} ∪ {(vi , vi′ )|1 ≤ i ≤ n}. The following two equivalent formulas can be found on page 42 in [1]:  2n−1  1 Y πj j t(M (n)) = 3 − (−1) − 2 cos , 2n j=1 n (4.5) √ i √ nh (2 + 3)n + (2 − 3)n + n. 2 (4.6) t(M (n)) = Now we use Theorem 2.1 to prove the following formula:  n  nY (2j − 1)π t(M (n)) = 4 − 2 cos . 2 j=1 n (4.7) Let f : V (M (n)) −→ V (M (n)) satisfy f (vi ) = vi′ and f (vi′ ) = vi for 1 ≤ i ≤ n. Obviously, f is an involution of M (n). By the definitions of M (n)R and M (n)L , M (n)R is a cycle with n vertices (see Figure 3(c)), and M (n)L is illustrated in Figure 3(b), where the weight of edge (v1 , vn ) equals -1, the weights of the two edges (v1 , u) and (vn , u) are 4 and the weights of the n − 2 edges (vi , u) for 2 ≤ i ≤ n − 1 are 2. Note that the submatrix obtained from the Laplacian matrix of M (n)L by deleting the row and column corresponding to vertex  0 1 0 ···   1 0 1 ···    0 1 0 ··· A= .. .. ..  ..  . . . .   0 0 0 ···  −1 0 0 · · · u equals 4In − A, where  0 −1  0 0    0 0  . .. ..   . .   0 1   1 0 n×n Now, the eigenvalues of A are 2 cos (2j−1)π for j = 1, 2, . . . , n. Hence, by the n Matrix-Tree Theorem, we have t(M (n)L ) = n  Y j=1 4 − 2 cos (2j − 1)π n  . By Theorem 2.1, we have  n  (2j − 1)π 1 nY 4 − 2 cos . t(M (n)) = t(M (n)L )t(M (n)R ) = 2 2 j=1 n Hence we have finished the proof of (4.7). (4.8) ENUMERATING SPANNING TREES OF GRAPHS WITH AN INVOLUTION 15 From (4.5), (4.6), and (4.7), we recover two interesting combinatorial identities as follows: 2 n = n−1 Y j=1 2jπ 2 − 2 cos n  ,  n  Y √ n √ n (2j − 1)π . 4 − 2 cos (2 + 3) + (2 − 3) + 2 = n j=1 4.4. The almost-join of two copies of a graph. Let G = (V (G), E(G)) be a graph with vertex set V (G) = {v1 , v2 , . . . , vs }. Take two copies of G which are denoted by G = (V (G), E(G)) and G′ = (V (G′ ), E(G′ )), where V (G′ ) = {v1′ , v2′ , . . . , vs′ } (hence φ : vi 7→ vi′ for 1 ≤ i ≤ s is an isomorphism between G and G′ ). The almost-join G⊙ of two copies of G is the graph with vertex set V (G) ∪ V (G′ ) and edge set E(G) ∪ E(G′ ) ∪ {(vi , vj′ )|vi ∈ V (G), vj′ ∈ V (G′ ), i 6= j}. Theorem 4.2. Let G⊙ be the almost-join of two copies of a graph G with s vertices defined as above. Then t(G⊙ ) = s−1 s−1 Y (s − 2 + µi )(s + µi ), s i=1 where 0 ≤ µ1 ≤ µ2 . . . ≤ µs−1 are the Laplacian eigenvalues of G. Proof Let f : V (G⊙ ) −→ V (G⊙ ) satisfy f (vi ) = vi′ and f (vi′ ) = vi for 1 ≤ i ≤ s. Obviously, f is an involution of G⊙ satisfying VC = ∅. By the definition of G⊙ L, ⊙ G⊙ L is the weighted graph with vertex set V (GL ) = {v1 , v2 , . . . , vs } ∪ {u} obtained from G by the following two procedures: 1. Add a new vertex u to G. 2. Add edges (vi , u) with weight 2(s − 1) for 1 ≤ i ≤ s, and add edges (vi , vj ) with weight −1 for 1 ≤ i, j ≤ s, i 6= j. Hence the degree di (G∗L ) of vertex vi in G⊙ L equals 2(s − 1) − (s − 1) + di (G) = s − 1 + di (G) for 1 ≤ i ≤ s, where di (G) denotes the degree of vertex vi in G. This shows that the submatrix of the Laplacian matrix of G⊙ L by deleting the row and column corresponding to vertex u is (s − 1)Is + diag(d1 (G), d2 (G), . . . , ds (G)) − A + (Js − I) = (s − 2)Is + Js + L(G), where Js is the matrix of order s with each entry equal to one, L(G) is the Laplacian matrix of G. Hence we have t(G⊙ L ) = det[(s − 2)Is + Js + L(G)]. (4.9) Note that Js and L(G) are symmetric, and commute. The vector v = (1, 1, . . . , 1)T spans the image of Js but also the kernel of L(G), and other eigenvectors of L(G) 16 FUJI ZHANG AND WEIGEN YAN are orthogonal to v and thus lay in the kernel of L(G). Since the eigenvalues of Js are s, 0, . . . , 0, Js + L(G) has eigenvalues s, µ1 , . . . , µs−1 . By (4.9), we have t(G⊙ L ) = (2s − 2) s−1 Y i=1 (s − 2 + µi ). (4.10) ⊙ ′ ′ ′ By the definition of G⊙ R , GR is the graph with vertex set {v1 , v2 , . . . , vs } obtained from G′ by adding edges (vi′ , vj′ ) with weight one for 1 ≤ i, j ≤ s, i 6= j. Hence the Laplacian matrix of G⊙ R equals (s − 1)Is + diag(d1 (G), d2 (G), . . . , ds (G)) − A(G) − (Js − Is ) = sIs + L(G) − Js . Note that the eigenvalues of L(G) − Js are −s, µ1 , . . . , µs−1 . Hence, by Lemma 3.3, we have t(G⊙ R) = s−1 1Y (s + µi ). s i=1 (4.11) By Theorem 2.1, the theorem follows from (4.10) and (4.11). Remark 4.3. In our knowledge, the results in Theorems 4.1 and 4.2 are new. Acknowledgements We are grateful to the referees for providing many friendly and helpful revising suggestions. One of them also told us an algebraic method to prove the two combinatorial identities in Subsection 4.3. References [1] N. L. Biggs, Algebraic Graph Theory, 2nd edn, Cambridge, Cambridge University Press, 1993. [2] J. A. Bondy and U. S. R. Murty, Graph Theory with Applications, American Elsevier, New York, 1976. [3] S.-C. Chang and R. Shrock, Some exact results for spanning trees on lattices, J. Phys. A: Math. Gen. 39 (2006), 5653–5658. [4] Z. Chen, On k-pairable graphs, Discrete Math., 27 (2004), 11–15. [5] M. Ciucu, Enumeration of perfect matchings in graphs with reflective symmetry, J. Combin. Theory Ser. A, 77 (1997), 67–97. [6] M. Ciucu, W. G. Yan, and F. J. Zhang, The number of spanning trees of plane graphs with reflective symmetry, J. Combin. Theory Ser. A, 112 (2005), 105–116. [7] N. Graham, R. C. Entringer, and L. A. Székely, New tricks for old trees: Maps and the Pigeonhole principle, Amer Math. Monthly, 101 (1994), 664–667. [8] R. W. Kenyon, J. G. Propp and D. B. Wilson, Trees and matchings, Electron. J. Combin., 7 (2000), #R25. ENUMERATING SPANNING TREES OF GRAPHS WITH AN INVOLUTION 17 [9] E. C. Kirby, D. J. Klein, R. B. Mallion, P. Pollak, H. Sachs, A Theorem for Counting Spanning Trees in General Chemical Graphs and Its Particular Application to Toroidal Fullerenes, Croat. Chem. Acta, 77 (2004), 263–78. [10] G. Kirchhoff, Über die Auflösung der Gleichungen, auf welche man bei der Untersuchung der linearen Verteilung galvanischer Ströme geführt wird, Ann. Phys. Chem., 72 (1847), 497–508. [11] R. Shrock and F. Y. Wu, Spanning trees on graphs and lattices in d dimensions, J. Phys. A: Math. Gen. 33 (2000), 3881–3902. [12] W. G. Yan, F. J. Zhang, Enumeration of perfect matchings of graphs with reflective symmetry by Pfaffians, Adv. Appl. Math., 32(2004), 655–668. [13] W. G. Yan, F. J. Zhang, Enumeration of perfect matchings of a type of Cartesian products of graphs, Disrete Appl. Math., 154(2006), 145–157. [14] D. J. A. Welsh, Complexity: Knots, Colourings and Counting, London Math. Soc., Lecture Notes Series 186, Cambridge, Cambridge University Press, 1993. School of Mathematical Science, Xiamen University, Xiamen 361005, China E-mail address: [email protected] Corresponding author, School of Sciences, Jimei University, Xiamen 361021, China E-mail address: [email protected]