The M-matrix group inverse problem for distance-biregular graphs

In this work, we obtain the group inverse of the combinatorial Laplacian matrix of distance-biregular graphs. This expression can be obtained trough the so-called equilibrium measures for sets obtained by deleting a vertex. Moreover, we show that the two equilibrium arrays characterizing distance-biregular graphs can be expressed in terms of the mentioned equilibrium measures. As a consequence of the minimum principle, we provide a characterization of when the group inverse of the combinatorial Laplacian matrix of a distance-biregular graph is an M-matrix.


Introduction
One problem with the theory of distance-regular graphs is that it does not apply directly to the graphs of generalised polygons.Godsil and Shawe-Taylor [24, ] overcame this difficulty by introducing the class of distance-regularised graphs, a natural common generalisation.These graphs are shown to either be distance-regular or distance-biregular.This family includes the generalised polygons and other interesting graphs.Distance-biregular graphs, which were introduced by Delorme [20] in 1983, can be viewed as a bipartite variant of distance-regular graph: the graphs are bipartite and for each vertex there exists an intersection array depending on the stable component of the vertex.Thus such graphs are to distance-regular graphs as bipartite regular graphs are to regular graphs.They also are to non-symmetric association schemes as distance-regular graphs are to symmetric association schemes.Since their introduction, distance-biregular graphs have received quite some attention, see [1,18,19,21,23,25,31] or [14,Chapter 4] for an overview.
In the first part of this paper we obtain the group inverse of the combinatorial Laplacian matrix of distance-biregular graphs.The group inverse matrix can be seen in the framework of discrete potential theory as the Green's functions associated with the Laplacian operator and it can be used to deal with diffusion-type problems on graphs, such as chip-firing, load balancing, and discrete Markov chains.For some graph classes, the group inverse is known.

Preliminaries
The triple Γ = (V, E, c) denotes a finite network; that is, a finite connected graph without loops or multiple edges, with vertex set V , whose cardinality equals n ≥ 2, and edge set E, in which each edge {x, y} has been assigned a conductance c(x, y) > 0. The conductance can be considered as a symmetric function c : V × V −→ [0, +∞) such that c(x, x) = 0 for any x ∈ V and moreover, x ∼ y, that is vertex x is adjacent to vertex y, iff c(x, y) > 0. We define the degree function k as k(x) = y∈V c(x, y) for each x ∈ V .The usual distance from vertex x to vertex y is denoted by d(x, y) and D = max{d(x, y) : x, y ∈ V } stands for the diameter of Γ.We denote as Γ i (x) the set of vertices at distance i from vertex x, Γ i (x) = {y : d(x, y) = i}, 0 ≤ i ≤ D and define k i (x) = Γ i (x) .Then, is the cardinal of the i-ball centered at x.The complement of Γ is defined as the graph Γ on the same vertices such that two vertices are adjacent iff they are not adjacent in Γ; that is x ∼ y in Γ iff c(x, y) = 0.More generally, for any i = 1, . . ., D, we denote by Γ i the graph whose vertices are those of Γ and in which two vertices are adjacent iff they are at distance i in Γ. Therefore for any x ∈ V , Γ i (x) is the set of adjacent vertices to x in Γ i .Clearly Γ 1 is the graph subjacent to the network Γ and Γ 2 = Γ when D = 2.
The set of real-valued functions on V is denoted by C(V ).When necessary, we identify the functions in C(V ) with vectors in R |V | and the endomorphisms of C(V ) with |V |-order square matrices.
The combinatorial Laplacian or simply the Laplacian of the graph Γ is the endomorphism of C(V ) that assigns to each u ∈ C(V ) the function It is well-known that L is a positive semi-definite self-adjoint operator and has 0 as its lowest eigenvalue whose associated eigenfunctions are constant.So, L can be interpreted as an irreducible, symmetric, diagonally dominant and singular M -matrix, that in the sequel will be denoted as L. Therefore, the Poisson equation and, when this happens, there exists a unique solution u ∈ C(V ) such that x∈V u(x) = 0, see [8].
The Green operator is the linear operator G : It is easy to prove that G is a positive semi-definite self-adjoint operator and has 0 as its lowest eigenvalue whose associated eigenfunctions are constant.Moreover, if P denotes the projection on the subspace of constant functions then, In addition, we define the Green function as G : where ε y stands for the Dirac function at y. Therefore, interpreting G, or G, as a matrix it is nothing else but L # the group inverse inverse of L, that coincides with its Moore-Penrose inverse.In consequence, L # is a M -matrix iff L # (x, y) ≤ 0 for any x, y ∈ V with x = y and then L # can be identified with the combinatorial Laplacian matrix of a new connected network with the same vertex set, that we denote by Γ # .From now on we will say that a network Γ has the M -property iff L # is an M -matrix; that is, if L provides an answer to Question 1.
In [8] it was proved that for any y ∈ V , there exists a unique ν y ∈ C(V ) such that ν y (y) = 0, ν y (x) > 0 for any x = y and satisfying We call ν y the equilibrium measure of V \ {y} and then we define capacity as the function cap ∈ C(V ) given by cap(y) = x∈V ν y (x).
Following the ideas in [8,12,34], we define, for any y ∈ V , the equilibrium array for y as the set {ν y (x) : x ∈ V } of different values taken by the equilibrium measure of y, and we consider the length of the equilibrium array to be ℓ(y) = ν y (x) : x ∈ V \ {y} .Since Γ is connected and n ≥ 2, we obtain that ℓ(y) ≥ 1 for any y ∈ V .On the other hand, since 0 = ν y (y) we obtain that {ν y (x) : x ∈ V } = {q i (y) : i = 0, . . ., ℓ(y)}, where 0 = q 0 (y) < q 1 (y) < • • • < q ℓ(y) (y).In addition, given y ∈ V for any i = 0, . . ., ℓ(y), we define m i (y) = {x ∈ V : ν y (x) = q i (y)} .Clearly, for any y ∈ V we have that In [11,Proposition 3.12] it was shown that, for any y ∈ V , the equilibrium measure (and hence the equilibrium array) reflects the graph depth from y, since and hence i j=0 for any 0 ≤ i ≤ D. In particular, (3) implies that if ν y (x) = q 1 (y) then x ∼ y; that is, that the minimum values of the equilibrium measure for y are attained at vertices adjacent to y (in fact this a formulation of the so-called minimum principle).
In general, when d(x, y) = i, Property (3) only assures that ν y (x) ≥ q i (y), but the inequality can be strict.In particular the length of some equilibrium arrays could be greater than D.
Example 2. To illustrate the above statements, consider the complete graph K 3 with vertex set V = {x 1 , x 2 , x 3 } and conductances c 1 = c(x 1 , x 2 ), c 2 = c(x 2 , x 3 ) and c 3 = c(x 3 , x 1 ).Then, The group inverse of the Laplacian matrix and the equilibrium measures provide an equivalent information about the network structure, since the expression of L # can be obtained from equilibrium measures and conversely.Specifically, see [8, Proposition 3.9], the group inverse L # is given by and this equality also implies that cap(y) = n 2 L # (y, y) and that In addition, the symmetry of the group inverse leads to the following relation for the equilibrium measures From ( 4) the minimum principle states that a network Γ has the M -property iff for any y ∈ V cap(y) ≤ nν y (x) for any x ∼ y, see [12,Theorem 1].In this case, Γ is a subgraph of the subjacent graph of Γ # .In fact, to achieve the M -property it is sufficient to satisfy that for any y ∈ V .Since this inequality trivially holds when ℓ(y) = 1, and assuming the common agreement that empty sum equals 0, we have that Γ has the M -property iff for any y ∈ V .Therefore, when ℓ(y) = 1 for any y ∈ V , then Γ is a complete network and moreover satisfies the M -property.As Example 2 shows, a complete network does not necessarily satisfy the M -property: , then K 3 does not satisfy the M -property.

Group inverse for distance-biregular graphs
We say that the graph there are numbers k 0 and k 1 such that each vertex in V 0 has k 0 neighbors and each vertex in V 1 has k 1 neighbors.In this case, we define Moreover, for ℓ = 0, 1, we denote by l = 1 − ℓ.In the sequel without loss of generality we always suppose that 1 A connected graph Γ is a distance-biregular graph if Γ is semiregular and for any two vertices x and y at distance i, the numbers |Γ i−1 (x) ∩ Γ 1 (y)| and |Γ i+1 (x) ∩ Γ 1 (y)| only depend on i and on the stable set where x is.
Examples of distance-biregular graphs are the subdivision graph of minimal (k, g)-cages.In particular, the subdivision graph of the Petersen graph is a distance-biregular graph.Also, any bipartite distance-regular graph is a distance-biregular graph with k 0 = k 1 .
For x ∈ V ℓ , ℓ = 0, 1, we define the intersection numbers by c ℓ,i 1 and more generally for any i ∈ {0, . . ., D ℓ } the following holds Therefore, a distance-biregular graph has a double intersection array which will be denoted by (vii) For ℓ = 0, 1, if i + j is even and i + j ≤ D ℓ , then c ℓ,i ≤ b ℓ,j .
The properties (ii), (iii) and (iv) imply that for ℓ = 0, 1, the intersection numbers {c ℓ,i , b ℓ,i } are determined by the intersection numbers {cl ,i , bl ,i }.In particular both sequences are the same iff k 0 = k 1 and in this case Γ is a (bipartite) distance-regular graph.
Lemma 4. [1, Corollary 2.11] Let Γ be a distance-bipartite regular graph.We can assume w.l.o.g. that one of the following holds 1. D 0 = D 1 and k 0 = k 1 ; so Γ is a bipartite distance-regular graph.

D
We display some preliminary results about the intersection parameters of a distancebiregular graphs, whose proofs are omitted since they follow trivially from [1].
The result provides an explicit expression of the equilibrium measure for sets V \ {y}, ∀y ∈ V of distance-biregular graphs.Proposition 6.Let Γ be a distance-biregular graph with V = V 0 ∪ V 1 .Then, for any ℓ = 0, 1, there exists an array q ℓ of length D ℓ such that if x ∈ V ℓ , for any y ∈ V it holds ν x (y) = q ℓ,m ⇐⇒ d(x, y) = m, m = 0, . . ., D ℓ .

Moreover,
In particular, Proof.Take x ∈ V ℓ with ℓ = 0, 1. Assume that the value ν x (y) depends only on the distance from x to y, that is, there exists q ℓ,i , i = 1, . . ., D ℓ such that ν x (y) = q ℓ,i ⇐⇒ d(x, y) = i.Moreover, we define q ℓ,D ℓ +1 = 0. Note that, since the equilibrium system Lν x (y) = 1 for all y ∈ V \ {x} has a unique solution, then if with our hypothesis we can solve the system, such solution must correspond to the equilibrium measure ν x (y) = q ℓ,i .
In our case, Lν x (y) = 1 for all y ∈ V \ {x} is equivalent to the system where ℓ = 0, 1. Multiplying by k ℓ,i , we obtain Observing that γ ℓ,D ℓ = 0, then summing up Finally, since q ℓ,0 = 0, it follows The expression for q ℓ,m in terms of c ℓ,i follows from Lemma 3 (ii).
Corollary 7. Let Γ be a distance-biregular graph with y ∈ V ℓ and x ∈ V l, ℓ, l = 0, 1.Then, Proof.From (6) we know that ν y (x) − ν x (y) = 1 n cap(y) − cap(x) .On the other hand, cap(y) = cap(z) for any z ∈ V ℓ and cap(x) = cap(w) for any w ∈ V l.So, for ℓ = l, we can choose z ∈ V ℓ and w ∈ V l such that d(z, w) = 1, then If ℓ = l, the result trivially holds.
As an straightforward application of Proposition 6 we can find the intersection array of a distance-biregular graph in terms of the equilibrium arrays, analogously as it was done in [11, Proposition 4.5] for distance-regular graphs.Proposition 8. Let Γ be a distance-biregular graph with equilibrium arrays q ℓ,i for ℓ = 0, 1 and i = 0, . . ., D ℓ .Then, for any i = 0, . . ., D ℓ − 1, it holds The computation of the equilibrium measure is usually done using linear programming [11].In this regard, Proposition 8 provides a tool to calculate the intersection arrays of a distance-biregular graph solving one linear system.
Another application of the equilibrium measure concerns the estimation of the effective resistance of a resistive electrical network.As a consequence of Proposition 6, we show the the effective resistance of distance-biregular graphs.
Corollary 9. Let Γ be a distance-biregular graph with y ∈ V ℓ and x ∈ V l, ℓ, l = 0, 1.Then, the effective resistance bewteen x, y is Proof.The result follows from Corollary 7 taking into account that R(x, y) = ν x (y) + ν y (x) n , see [9].
The next results shows the group inverse of the Laplacian of a distance-biregular graphs in terms of the intersection arrays.
Theorem 10.Let Γ be a distance-biregular graph.Then, for each y ∈ V ℓ with ℓ = 0, 1, the group inverse of L is given by Proof.From (4), we know that L # (x, y) = 1 n 2 (cap(y) − nν y (x)).Take y ∈ V ℓ with ℓ = 0, 1.Now, using Proposition 6, Therefore, Remark 11.Observe that, since the intersection numbers of a distance-biregular graph are related, from Theorem 10, the expression of the group inverse is equivalent to Example 12.As an application of Theorem 10 we obtain the group inverse of the Laplacian of a complete bipartite graph using its parameters: Observe that for a complete bipartite graph, it holds that L # is always an M -matrix.The above expression is valid when D 0 = D 1 = 1, and D 0 = 1 and D 1 = 2; that is, for the star graph.

Distance-biregular graphs with the M-property
In this section, we answer Question 1 for distance-biregular graphs, completing, together with the known results for distance-regular graphs [12], the characterization of when the group inverse of the combinatorial Laplacian matrix of a distance-regularised graph is an M -matrix.
Proposition 13.Let Γ be a distance-biregular graph.Then, Γ has the M -property if and only if, it holds Proof.We know that a graph Γ satisfies the M -property if and only if the entries of L # (x, y) ≤ 0 for all x ∼ y.The result follows from using that k 0,0 = 1, b 0,0 = k 0 , and Remark 14.The condition from Proposition 13 is equivalent to Using Proposition 13 we can also obtain the following necessary condition for a distancebiregular graph having the M -property.Proof.Since D 1 ≥ D 0 ≥ 2, from Proposition 13 we obtain that Now, observing that we get and since n−1 > 0, the result follows.
Note that the inequality n < 2k 1 + k 0 turns out to be a strong restriction for a distancebiregular graph to have the M -property.Observe that such condition implies that the distance-biregular graph needs to be quite dense.
The following result generalizes the above observation by showing that only distancebiregular graphs with small D ℓ can satisfy the M -property.A related result appeared in [12,Proposition 5], where it was shown that the diameter of a distance-regular graphs with the M -property must be at most 3. Proof.By means of a contradiction, assume D 0 , D 1 ≥ 4.Then, We can assume that k 0 > k 1 , since otherwise Γ is a bipartite distance-regular graphs and hence D 0 = D 1 ≤ 3, see [12,Proposition 5].Then, On the other hand, since b 0,1 ≥ c 0,3 and b 0,2 ≥ c 0,2 , we obtain that Finally, from Corollary 15 it follows that n + 1 As an application of Proposition 16, we classify distance-biregular graphs having the M -property.We follow the notation from [1].
Case 1: D 0 = D 1 = 1.This case corresponds to a digon that is a distance-regular and has the M -property.
Next, using the condition in Proposition 13 with r = k 0 , k = k 1 , we obtain a necessary and sufficient condition for a distance-biregular graph with D 0 = 3, D 1 = 4 to have the M -property.
Proof.We use Proposition 13 to obtain: Keeping in mind that After performing some simplifications on the first inequality, the desired result follows.
Finally, we study the M -property for some classes of distance-biregular graphs with D 0 = 3 and D 1 = 4.
Example 19.Consider the point-line incident graph of the affine plane A(2, n) of order n, whose intersection array is n + 1; 1, 1, n n; 1, 1, n, n .
It is easy to check that it does not verify the inequality in Proposition 13 and hence it does not verify the M -property.which does not hold the condition from Proposition 13, and thus does not have the Mproperty.In fact, we can find the group inverse.We denote L # ℓ,j = L # (x, y) when y ∈ V ℓ and d(x, y) = j.Then, For the classification of existing quasi-symmetric 2-designs, see [33,Table 48.25].We should note that for the existing quasi-symmetric 2-designs with x = 0, none passes the condition from Proposition 18.
The above discussion extends the results in [12] and completes the classification of distance-regularised graphs that have the M -property.There, it was shown that if there are distance-regular graphs with valency k ≥ 3 and diameter D ≥ 2 having the M -property, then they have at most 3k vertices and D ≤ 3. Also in [12], it was conjectured that there is no primitive distance-regular graph with diameter 3 having the M -property.This conjecture was shown to be true except possibly for finitely many primitive distance-regular graphs [30,Theorem 1].
In view of the above results, we conclude this paper with the following conjecture.
Conjecture 21.There are not point-line incidence graphs of a 2-quasi-symmetric design x = 0 that have the M -property.

Proposition 16 .
If Γ is a distance-biregular graph with the M -property, then D 1 ≤ 4 and D 0 ≤ 3.

Proposition 18 .
A distance-biregular graph with diameters D 0 = 3, D 1 = 4 has the Mproperty if and only if
1 and moreover the following relationships hold, see the Lemmas 2.1-2.8 in [1, Section 2.1] and the references therein.Lemma 3. If Γ is a distance-biregular graphs with intersection arrays k ℓ