A combinatorial expression for the group inverse of symmetric M-matrices

Abstract: By using combinatorial techniques, we obtain an extension of the matrix-tree theorem for general symmetric M-matrices with no restrictions, this means that we do not have to assume the diagonally dominance hypothesis. We express the group inverse of a symmetric M–matrix in terms of the weight of spanning rooted forests. In fact, we give a combinatorial expression for both the determinant of the considered matrix and the determinant of any submatrix obtained by deleting a row and a column. Moreover, the singular case is obtained as a limit case when certain parameter goes to zero. In particular, we recover some known results regarding trees. As examples that illustrate our results we give the expressions for the Group inverse of any symmetric M-matrix of order two and three. We also consider the case of the cycle C4 an example of a non-contractible situation topologically di erent from a tree. Finally, we obtain some relations between combinatorial numbers, such as Horadam, Fibonacci or Pell numbers and the number of spanning rooted trees on a path.


Introduction
The classical matrix-tree theorem, whose rst version was proved by G. Kirchho in 1847, relates the principal minors of the Laplacian matrix of a networkΓ, with the total weight of spanning trees of Γ. Therefore, it refers to singular, symmetric and diagonally dominant M-matrices. Since then, many generalizations and di erent proofs have been considered (see for instance [1] and the references therein). In particular, the matrix-tree theorem has been extended to the case of non-singular, symmetric, α-diagonally dominant M-matrices, see [12,16,26,27]. By α-diagonally dominant M-matrix, we understand those diagonally dominant M-matrices that have equal excess, α, in the diagonal. Hence, the value of the determinant is expressed as a polynomial in α whose coe cients are given by the weight of spanning rooted forest. In [21], the case of non-symmetric and (non-singular) Laplacian matrices is consider for the case of complex weights. In general, it is not an easy task to compute the number of spanning trees together with their weights in the weighted case, see [25] for some explicit expression in the case that the conductances are tensorial product and the underlying graph is structured. In [28] we found expressions for enumerating spanning trees in bipartite graphs, line graphs, generalized line graphs, middle graphs, total graphs, generalized join graphs and vertex-weighted graphs.
In this work we obtain an extension of the matrix-tree theorem for general symmetric M-matrices; which means that we remove the diagonally dominance hypothesis. The techniques we use are of combinatorial nature. Speci cally, to prove our results we use an extension of the undirected version of the all-minors theorem by S. Chaiken [11], see also [8] for a beautiful application. We give a combinatorial expression for both the determinant of the considered matrix and the determinant of any submatrix obtained by deleting a row and a column. Thus, we express the group inverse of a symmetric M-matrix in terms of the weight of spanning rooted forests. Moreover, the singular case is obtained as a limit case when the parameter goes to zero and we recover some known result regarding trees.
Given a nite set V, the set of real valued functions on V is denoted by C(V). The standard inner product on C(V) is denoted by ·, · and hence, if u, v ∈ C(V), then u, v = x∈V u(x) v(x). For any x ∈ V, εx ∈ C(V) stands for the Dirac function at x and is the function de ned by (x) = , for any x ∈ V. On the other hand, ω ∈ C(V) is called a weight if it satis es that ω(x) > for any x ∈ V and moreover ω, ω = n = |V|. The set of weights on V is denoted by Ω(V).
The triple Γ = (V , E, c) denotes a nite network; that is, a nite connected graph without loops nor multiple edges, with vertex set V, whose cardinality equals n, and edge set E, in which each edge {x, y} has been assigned a conductance c(x, y) > . So, the conductance can be considered as a symmetric function c : V × V −→ [ , +∞) such that c(x, x) = for any x ∈ V and moreover, vertex x is adjacent to vertex y i c(x, y) > . If we label the set of vertices, then functions can be identi ed with vectors in R n and endomorphisms of C(V) with matrices of order n.
The combinatorial Laplacian or simply the Laplacian of the network Γ is the endomorphism of C(V) that assigns to each u ∈ C(V) the function The matrix associated with L is always a singular symmetric M-matrix. Recall that a symmetric matrix M is called a M-matrix if its o diagonal entries are non-positive, and its eigenvalues are non-negative. Moreover, a matrix M is called irreducible if it is not similar via a permutation to a block upper triangular matrix. Therefore, a symmetric matrix is irreducible if and only if its associated graph is connected.
Given q ∈ C(V), the Schrödinger operator on Γ with potential q is the endomorphism of C(V) that assigns to each u ∈ C(V) the function Lq(u) = L(u) + qu, where qu ∈ C(V) is de ned as (qu)(x) = q(x)u(x); see for instance [2,6].
Given a weight ω ∈ Ω(V), we called Doob-potential determined by ω the function given by qω = −ω − L(ω). It is well-known that any Schrödinger operator is self-adjoint and moreover it is positive semide nite i there exist ω ∈ Ω(V) and λ ≥ such that q = qω + λ; see [2]. In addition, Lq is singular i λ = , in which case Lq ω (v), v = i v = aω, a ∈ R. In any case, λ is the lowest eigenvalue of Lq and its associated eigenfunctions are multiple of ω. In [4], some of the authors proved that any irreducible, symmetric M-matrix can be identi ed with a positive semi-de nite Schrödinger operator on a network.
In this work we x ω ∈ Ω(V) and for each λ ≥ , we consider the positive semi-de nite Schrödinger operator Lq, where q = qω + λ. Then, Lq is an automorphism on ω ⊥ , whose inverse is called (orthogonal ) Green operator and is denoted by Gq. We can extend the Green operator to C(V) by assigning to any f ∈ C(V), the unique u ∈ C(V) such that Lq(u) = f − n ω, f ω and u, ω = . The Green operator is self-adjoint, singular and satis es that Gq(ω) = , see [4].
The function Gq : V × V −→ R, de ned as Gq(x, y) = Gq(εy)(x), for any x, y ∈ V, is called Green function. Moreover, for λ = , the matrix associated with Gq is the Moore-Penrose inverse of Lq (that in this case coincides with the group inverse); whereas, for λ > , we have that L − q = Gq + nλ ω ⊗ ω. Through the manuscript, we use a common technique in the context of electrical networks and Markov chains, see [17,18]. The authors considered this technique in [4] in order to generalize the Fiedler characterization of irreducible, symmetric and diagonally dominant M-matrices as resistive inverses, see [20], to all irreducible and symmetric M-matrices or equivalently, to all positive semi-de nite Schrödinger operators. Given a positive-de nite Schrödinger operator on Γ, Lq, the method consists in embedding the given network into a suitable host network. The new network is constructed by adding a new vertex, that represents a grounded vertex or an absorbing state in the context of Markov chains. The new vertex,x, is joined with each vertex in the original network through a new edge whose conductance is the diagonal excess of the original M-matrix after the application of the Doob transform; see Figure 1.
The next result, whose proof can be found in [4,Proposition 2.8], establishes the relationship between the original Schrödinger operator Lq and a new semide nite Schrödinger operator on Γ λ,ω .

Matrix Tree Theorem for symmetric M-matrices
In this section we aim to present the matrix tree-theorem for positive de nite Schrödinger operators and to get the relation between the number of rooted forests and the Green function of the network. Moreover, taking limits when λ goes to we get the matrix tree-theorem for positive semi-de nite Schrödinger operators. If T is a network such that for each pair x and y of vertices, there is exactly one path joining x and y, then T is called a weighted tree. Therefore, T is a weighted tree i it has no cycles. Given Γ a connected network, we say that T is a spanning tree of Γ if both networks have the same set of vertices and each edge of T is also an edge of Γ with the same weight. A forest is a non-necessarily connected network without cycles. A rooted forest is a forest with one vertex marked as a root in each connected component. Moreover a k-rooted forest is a forest with k components and a vertex marked as a root in each connected component. The set of vertices and the set of edges of a forest F, will be denoted by V(F) and E(F), respectively. The weight of a forest F is de ned as the product of all weights of the edges of F taking into account the weights of the vertices given by ω; i.e., where e = {x, y} ∈ E(F) and w(e) = c(x, y)ω(x)ω(y) and we take the convention that the empty product equals . We denote by F = F(Γ) the set of rooted spanning forests of Γ and F k = F k (Γ) the set of k-rooted spanning forest of Γ. Moreover, Fx = Fx(Γ) denotes the set of rooted spanning forests of Γ such that x is a root and F k,x = F k,x (Γ) denotes the set of k-rooted spanning forests of Γ such that x is a root. Finally, we denote by Fx,y = Fx,y(Γ) the set of rooted spanning forests of Γ such that x is a root and y belongs to the same component as x and F k,x,y = F k,x,y (Γ) the set of k-rooted spanning forest of Γ such that x is a root and y belongs to the same component as x. On the other hand, F {y} k,x denotes the set of rooted spanning forest of Γ such that x is a root and y does not belong to the same component as x. If F ∈ F k we denote by r(F) the set of its roots whereas if F ∈ F k,x we denote by r * (F) the set of roots di erent from x. In order to state the matrix notation we need to label the vertices and the edges of the network. So, let Γ = (V , E, c) be a network and suppose that V = { , . . . , n} and E = {e , . . . , em}. Assume that each edge of Γ is assigned an orientation, which is arbitrary but xed. Then, e = (i, j), for = , . . . , m and i, j = , . . . , n, where i is the tail and j is the head of e . We de ne the weight of and edge as w(e ) = c(i, j)ω(i)ω(j). Then, the weighted incidence matrix Bω = (b i ) ∈ Mm×n is de ned as and we denote by W(c, ω) ∈ Mm, the diagonal matrix whose diagonal elements are given by w(e ). Using this notation Moreover, the matrix associated with Lq is given by If we denote by L λ,ω q , the matrix associated with the Schödinger operator in the host network, Equation (2) can be expressed as where L λ,ω q (n + |n + ) is the submatrix of L λ,ω q obtained by deleting the (n + )-th row and column. Moreover, the relation between weighted incidence matrices is the following (c, ω) ∈ M m+n (R) and j denotes the all ones vector. Therefore, if B λ,ω ω (n + ) denotes the submatrix of B λ,ω ω obtained by deleting the (n + )-th column, we get Note, that the above matrices are similar to the ones introduced in [16] in relation with the normalized Laplacian. Notice that with our interpretation these matrices acquire a real meanig.
Observe that ifF is a spanning tree on Γ λ,ω with vertex n + as a root and k edges of type {n + , i}, the forest generated in Γ, F, has k connected components and each i belongs to a di erent component and hence it can be considered as a root of the forest, see Figure 4. With this interpretation ofF, we can state some generalization of the matrix-tree Theorem. This extension considers not only weights in the edges, as in the previous works, but also weights on the vertices. Moreover, the result applies to general irreducible symmetric M-matrices. Although the proof is based on the all minor tree Theorem due to Chaiken, [11], we follow here the clearer terminology introduced in [8]. Theorem 2.1. [8,11] where the sum is taken over the set of all J-admissible forest F of Γ and • means the composition of two permutations.
The cases in which = , have an special meaning for our pourposes.
Observe that det(Lq) = det(Lq ω + λI), and hence an must equal and a n− = tr(Lq ω ) = x,y∈V . This values can also be deduced from the fact that there is only one forest with n connected components and there are m rooted forest with n − components. Moreover, notice that the above result is also valid for λ = , since in this case det(Lq ω ) = .
Proposition 2.3. Let λ > and ω ∈ Ω(V). Then, for any t, s ∈ V, In particular, when t = s, Proof. From Equation (4) where F is a (k + )-rooted spanning forest in Γ with t ∈ r(F) as set of roots and t, s belong to the same connected component. When t = s, the formula follows straightforward. Observe that the above result for the speci c case of the normalized Laplacian was obtained by Chung and Zhao in [16,Theorem 3 ], using an intricate proof based on Cauchy-Binet Theorem. Chebotarev et alt., in a serie of papers, also obtained a similar result for the case of weighted (di)graphs for the matrix L + λI, [12,13].
The following result represents a full weighted version of the classical matrix tree Theorem, [7,22]. It can be obtained either directly from Chaiken's Theorem or taking limit when λ goes to in the previous result.

Corollary 2.4. Let ω ∈ Ω(V). Then, for any t, s ∈ V,
where F runs on the set of spanning trees of Γ.

The Group inverse of symmetric M-matrices
Using the matrix-tree theorems in the preceding section we can write down a combinatorial expression for the group inverse of any irreducible symmetric M-matrix. To do this we consider the associated Schrödinger operator, Lq, and then we get its Green function in terms of the weights of rooted spanning forests in Γ.

In particular, nb (t, s) = a , and hence b is independent of the variables t, s. Moreover, for λ =
Proof. Applying Theorems 2.3 and 2.2 to the adjoint formula for the inverse of a non-singular matrix, we get that Hence, we get that The result for λ = follows by taking limit in the above expression.
The following result contains the particular case in which ω is the constant weight. So, the operator is Lq = L + λI where L is the combinatorial Laplacian of a network. This result coincides with the ones obtained in [13,26,27]. In particular, for λ = , For graphs we obtain the expression for the Group inverse of the combinatorial Laplacian L that has been widely considered in the literature, [13,16]. It means that for λ = , In the following example we show how to compute the combinatorial coe cients a k and b k (x t , xs) in the case of a star on n vertices with center at x and conductances c i , i = , . . . , n − , see Figure 5. In [9, Corollary 5.3] the expression for G λ (x t , xs) was obtained by the authors using di erent techniques.  Moreover, for any k = , . . . , n, we have To end this section we consider the computation of some small cases that will show the spirit of the results. The Group inverse for K . Observe that this case encompasses the case of any × symmetric M-matrix.
x )} and c = c(x , x ) > . Moreover, we consider the weight ω = ω(x ), ω = ω(x ), where ω + ω = , the parameter λ ≥ and q = qω + λ. In this case, For λ > , we can compute directly the inverse of Lq, obtaining On the other hand, we get that In Figure 6 we can see the existing trees. Therefore, keeping in mind that a + λa = c ω ω + λ, we get that Figure 6: The network K (left) and the collection of -rooted forests for K (right). Moreover, , , Finally, for λ = The Group inverse for K . Now, we consider any symmetric M-matrix of order 3. For that, Moreover, we consider the weight ω = ω(x ), ω = ω(x ), ω = ω(x ), where k= ω i = , w i = c i ω i ω i+ , the parameter λ ≥ and q = qω + λ. In this case, Next, we compute the coe cients b and a involved in the formula of the group inverse. We also represent the trees or forests of any order in Figures 7-9. Figure 7: The network K (left) and the collection of spanning trees for K (right).
x 1 To compute coe cients b k (x i , x j ) it is enough to count the k-rooted forests displayed at Figures 8 and 9 with x i as a root and x j at the same component as x 1 x 2 x 3 Figure 9: The collection of -rooted forests for K .
Applying Theorems 2.3 and 2.2, we get that the inverse of Lq for K is Moreover, from Proposition 4.1 we get that the expression for the Green function for K and λ ≥ is In particular, for λ = , we get that

The group inverse for trees
Consider now a weighted tree with constant weights on the vertex set, T. For any k ≥ and any set of edges E k ⊂ E(T) satisfying |E k | = k − , we let T \ E k denote the graph obtained from T by deleting the edges of E k . Observe that, T \ E k is a forest that has k connected components, that will be denoted by C i (E k ), i = , . . . , k.
For each set E k ∈ E(T) and a vertex t ∈ T, we denote C t i (E k ), i = , . . . , k − , the set of vertices in the i-th connected component of T \ E k that does not contain vertex t. Moreover, for any pair of vertices t, s ∈ V(T), we denote by E t→s k ⊂ E(T) a set of k − edges that are not in the unique path from t to s in T, P t,s . Finally, denote by w(E k ) = e∈E k c(e). .
In particular, for λ = we have that Proof. It is enough to observe that for any t, s ∈ Observe that the result in the above Corollary for λ = , coincides with the one obtained in [23, Corollary 7.2.6], see also [24,Lemma 2.14].
In particular, if we consider a path on n vertices and keeping in mind that |C t (e k )| = k for any e k whose initial vertex is k ≤ t − and |C t (e k )| = n − k for any e k whose initial vertex is k ≥ t, we recover the expression for the Green function obtained by the authors in [5] In particular, .
For the case of a path with constant conductances the authors obtained in [3] the inverse of L λ by using difference equation techniques and comparing both expressions we can state the following in some sense surprising identities. where Un(x) and Vn(x) are the second and third kind Chebyshev polynomials.
It is notable that di erent values of the parameter λ enable us to give more or less importance to the length of the di erent walks between vertices, [10,15]. Moreover, we can obtain some relations between combinatorial numbers and the number of spanning rooted trees. First, let us de ne the Horadam numbers Hn(r, s) as the solution of the following three term recurrence sequence H n+ (r, s) = rH n+ (r, s) + sHn(r, s), H (r, s) = , H (r, s) = , r, s ∈ Z.
See [19] for properties of Horadam numbers. In particular, it is well known that Hn( , ) = Fn, the n-th In particular for λ = , and , we get Observe that the case λ = gives that F n is the total number of spanning rooted forests of the path, see [14] for a di erent proof of this result. On the other hand, P n represents the number of rooted spanning forest taking into account a weight associated with the number of components of the forest, so the more components the greater the weight. In contrast, for the numbers J n the less components the greater the weight.

The Group inverse for C
In this section we consider a cycle of order , and hence we look for the inverse or the Green function of some especial class of symmetric M-matrix of order . Let Γ = (V , E, c),