Carmona Mejías, Ángeles

Encinas Bachiller, Andrés Marcos

Jiménez Jiménez, María José

English

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MEAN FIRST PASSAGE TIME FOR DISTANCE-BIREGULARGRAPHSMARÍA JOSÉ JIMÉNEZIn this presentation, we obtain the explicit expression for the Group inverse ofthe Laplacian matrix associated with distance–biregular graphs, ([2]). A bipartitegraph is called distance–biregular (DBR) if all the vertices of the same partiteset admit the same intersection array. So, this kind of graphs are characterizedby having two intersection arrays instead of one as in the case of distance-regulargraphs. Examples of this kind of graphs are complete bipartite graphs, subdivisiongraphs of minimal cages and some block designs, see [19]. As an application, weprovide the mean first passage time for DBR graphs as well as the Kemeny constant.The above expression will be given in terms of the so–called equilibrium measurefor a vertex {x}, see [6]. Finally, we provide some examples as star graphs.This is joint work with Á. Carmona and A.M. Encinas (UPC). Partially sup-ported by the Departament de Matemàtiques of UPC.1. IntroductionOne problem with the theory of distance-regular graphs is that it does not applydirectly to the graphs of generalised polygons. Godsil and Shawe–Taylor [19, ] over-came this difficulty by introducing the class of distance-regularised graphs, a naturalcommon generalisation. These graphs are shown to either be distance-regular ordistance-biregular. This family includes the generalised polygons and other inter-esting graphs. Distance-biregular graphs, which were introduced by Delorme [16]in 1983, can be viewed as a bipartite variant of distance-regular graph: the graphsare bipartite and for each vertex there exists an intersection array depending on thestable component of the vertex. Thus such graphs are to distance-regular graphs asbipartite regular graphs are to regular graphs. They also are to non-symmetric as-sociation schemes as distance-regular graphs are to symmetric association schemes.Since their introduction, distance-biregular graphs have received quite some atten-tion, see [1, 14, 15, 17, 18, 20, 24] or [9, Chapter 4] for an overview.First, we obtain the group inverse of the combinatorial Laplacian matrix ofdistance-biregular graphs, so the mean first passage time can be calculated in termsof that matrix. The group inverse matrix can be seen in the framework of discretepotential theory as the Green’s functions associated with the Laplacian operatorand 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, thegroup inverse is known. Instances of it are the work of Urakawa [25], Bendito etalt. [6, 7] or more recently the study of the Green function for forests by Chungand Zeng [13]. Other generalized inverses, such as the Moore-Penrose inverse, havebeen studied. For instance, the Moore-Penrose inverse of the incidence matrix ofseveral graphs has been investigated by Azami and Bapat [3, 4, 5]. Nevertheless,the problem of computing group inverses still remains wide open for most graphclasses. We show an explicit expression for the group inverse of the combinatorialLaplacian matrix of a distance-biregular graph in terms of its intersection numbers.This result, together with the group inverse of a distance-regular graph found by1Bendito, Carmona and Encinas [7], and independently, by Chung and Yau [12],completes the corresponding study for distance-regularised graphs.In matrix theory, the Laplacian matrix is known to be a symmetric M -matrix (asymmetric positive semi-definite matrix with non-positive off-diagonal elements).Nonnegative matrices and M -matrices have become a staple in contemporary linearalgebra, and they arise frequently in its applications. Such matrices are encounterednot only in matrix analysis, but also in stochastic processes, graph theory, electricalnetworks, and demographic models [22]. A fundamental problem related with M -matrices is the so-called inverse M -matrix problem, that consists in characterizingall non-negative matrices whose inverses are M -matrices. For singular matrices, theinverse problem was originally posted by Neumann, Poole and Werner as follows:Characterize all singular and irreducible M -matrices for which its group inverse isalso an M -matrix.This question has only been answered for specific matrix classes. In the graphsetting, this question has been solved for weighted trees by Kirkland and Neu-mann [23], and for distance-regular graphs by Bendito, Carmona and Encinas [8].In a more general setting, it has been investigated for nonnegative matrices hav-ing few eigenvalues by Kirkland and Neumann [22], for periodic and nonperiodicJacobi matrices by Chen, Kirkland and Neumann [11] and for general symmetricM -matrices whose underlying graphs are paths by Bendito, Carmona and Enci-nas [8] and Carmona, Encinas and Mitjana [10]. Recently, matrices whose groupinverses are M -matrices were investigated by Kalauch, Lavanya and Sivakumar [21].We answer the question for distance-biregular graphs, completing, together withthe known results for distance-regular graphs [8], the characterization of when thegroup inverse of the combinatorial Laplacian matrix of a distance–regularised graphis an M–matrix.2. Group inverse for distance-biregular graphsThe triple Γ = (V,E, c) denotes a finite network; that is, a finite connectedgraph without loops or multiple edges, with vertex set V , whose cardinality equalsn ≥ 2, and edge set E, in which each edge {x, y} has been assigned a conductancec(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 isvertex x is adjacent to vertex y, iff c(x, y) > 0. We define the degree function k ask(x) =∑y∈Vc(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 Γi(x)the set of vertices at distance i from vertex x, Γi(x) = {y : d(x, y) = i}, 0 ≤ i ≤ Dand define ki(x) =∣∣Γi(x)∣∣. Then,Bi(x) =i∑j=0kj(x)is the cardinal of the i-ball centered at x. The complement of Γ is defined as thegraph Γ on the same vertices such that two vertices are adjacent iff they are notadjacent in Γ; that is x ∼ y in Γ iff c(x, y) = 0. More generally, for any i = 1, . . . , D,we denote by Γi the graph whose vertices are those of Γ and in which two verticesare adjacent iff they are at distance i in Γ. Therefore for any x ∈ V , Γi(x) is theset of adjacent vertices to x in Γi. Clearly Γ1 is the graph subjacent to the networkΓ and Γ2 = Γ when D = 2.The set of real–valued functions on V is denoted by C(V ). When necessary, weidentify 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 en-domorphism of C(V ) that assigns to each u ∈ C(V ) the function(1) L(u)(x) =∑y∈Vc(x, y)(u(x)− u(y))= k(x)u(x)−∑y∈Vc(x, y)u(y), x ∈ V.It is well–known that L is a positive semi–definite self–adjoint operator and has0 as its lowest eigenvalue whose associated eigenfunctions are constant. So, L canbe interpreted as an irreducible, symmetric, diagonally dominant and singular M–matrix, that in the sequel will be denoted as L. Therefore, the Poisson equationL(u) = f on V has solution iff ∑x∈Vf(x) = 0and, when this happens, there exists a unique solution u ∈ C(V ) such that∑x∈Vu(x) =0, see [6].The Green operator is the linear operator G : C(V ) −→ C(V ) that assigns toany f ∈ C(V ) the unique solution of the Poisson equation L(u) = f − 1n∑x∈Vf(x)such that∑x∈Vu(x) = 0. It is easy to prove that G is a positive semi–definite self–adjoint operator and has 0 as its lowest eigenvalue whose associated eigenfunctionsare constant. Moreover, if P denotes the projection on the subspace of constantfunctions then,L ◦G = G ◦ L = I − P.In addition, we define the Green function as G : V × V −→ R given by G(x, y) =G(εy)(x), where εy 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, thatcoincides with its Moore-Penrose inverse. In consequence, L# is a M–matrix iffL#(x, y) ≤ 0 for any x, y ∈ V with x 6= y and then L# can be identified with thecombinatorial Laplacian matrix of a new connected network with the same vertexset, that we denote by Γ#.From now on we will say that a network Γ has the M–property iff L# is anM–matrix; that is, if L provides an answer our question.In [6] it was proved that for any y ∈ V , there exists a unique νy ∈ C(V ) suchthat νy(y) = 0, νy(x) > 0 for any x 6= y and satisfying(2) L(νy) = −nεy on V .We call νy the equilibrium measure of V \ {y} and then we define capacity as thefunction ∈ C(V ) given by (y) =∑x∈Vνy(x).Following the ideas in [6, 8, 25], we define, for any y ∈ V , the equilibrium arrayfor y as the set {νy(x) : x ∈ V } of different values taken by the equilibrium measureof y, and we consider the length of the equilibrium array to be `(y) =∣∣{νy(x) : x ∈V \ {y}}∣∣. Since Γ is connected and n ≥ 2, we obtain that `(y) ≥ 1 for any y ∈ V .On the other hand, since 0 = νy(y) we obtain that {νy(x) : x ∈ V } = {qi(y) : i =0, . . . , `(y)}, where 0 = q0(y) < q1(y) < · · · < q`(y)(y). In addition, given y ∈ V forany i = 0, . . . , `(y), we define mi(y) =∣∣{x ∈ V : νy(x) = qi(y)}∣∣. Clearly, for anyy ∈ V we have thatm0(y) = 1, n =`(y)∑i=0mi(y), and (y) =`(y)∑i=1mi(y)qi(y).The group inverse of the Laplacian matrix and the equilibrium measures providean equivalent information about the network structure, since the expression of #can be obtained from equilibrium measures and conversely. Specifically, see [6,Proposition 3.9], the group inverse L# is given by(3) L#(x, y) =1n2((y)− n νy(x)).In addition, the symmetry of the group inverse leads to the following relation forthe equilibrium measures(4) νy(x)− νx(y) = 1n((y)− (x))= n(#(y, y)−# (x, x)), x, y ∈ V.References[1] J.M. van den Akker, Distance-biregular graphs, MSc thesis Eindhoven University of Technol-ogy, 1990.[2] A. Abiad, Á. Carmona, A.M. Encinas and M.J. Jiménez. The M -matrix group inverse problemfor distance–biregular graphs. submitted, (2022).[3] A. Azimi and R.B. Bapat, The Moore-Penrose inverse of the incidence matrix of completemultipartite and bi-block graphs, Discrete Math. 342 (2019), 2393–2401.[4] A. Azimi, R.B. Bapat and E. Estaji, Moore-Penrose inverse of incidence matrix of graphs withcomplete and cycle blocks, Discrete Math. 342 (2019), 10–17.[5] A. Azimi and R.B. Bapat, Moore-Penrose inverse of the incidence matrix of a distance regulargraph, Linear Algebra Appl. 551 (2018) 92–103.[6] E. Bendito, Á. Carmona, A.M. Encinas. Solving boundary value problems on networks usingequilibrium measures. J. Func. Anal., 171: 155–176, (2000).[7] E. Bendito, A. Carmona, A.M. Encinas and M. Mitjana, Generalized inverses of symmetricM-matrices, Linear Algebra Appl. 432 (2010), 2438–2454.[8] E. Bendito, A. Carmona, A.M. Encinas and M. Mitjana, Distance-regular graphs having theM-property, Linear Multilinear Algebra 60 (2012), 225–240.[9] A.E. Brouwer, A.M. Cohen and A. Neumaier, Distance–regular graphs. Ergebnisse der Math-ematik und ihrer Grenzgebiete, vol. 18, Springer-Verlag, Berlin, 1989.[10] A. Carmona, A.M. Encinas, M. Mitjana, On the M -matrix inverse problem for singular andsymmetric Jacobi matrices, Electron. J. Linear Algebra 24 (2013), 237–254.[11] Y. Chen, S.J. Kirkland and M. Neumann, Group generalized inverses of M -matrices asso-ciated with periodic and nonperiodic Jacobi matrices, Linear Multilinear Algebra 39 (1995),325–340.[12] F. Chung and S.-T. Yau, Discrete Green’s Functions, J. Combin. Theory Ser A 91(1-2)(2000), 191–214.[13] F. Chung and J. Zeng, Forest formulas of discrete Green’s functions, arXiv:2109.01324.[14] B. Curtin, Bipartite Distance-regular Graphs, Part I, Graphs Combin. 15 (1999), 143–158.[15] B. Curtin, Bipartite Distance-regular Graphs, Part II, Graphs Combin. 15 (1999), 377–391.[16] C. Delorme, Régularité métrique forte, Rapport de Reserche No. 156, Univ. Paris Sud, Orsay(1983).[17] C. Delorme, Distance biregular bipartite graphs, European J. Combin. 15 (1994), 223–238.[18] M.A. Fiol, The spectral excess theorem for distance-biregular graphs, Electron. J. Combin.20(3) (2013), #P21.[19] C.D. Godsil, J.S. Taylor. Distance-regularised graphs are distance-regular or distance-biregular. J. Combin. Theo., series B, 43: 14–24, (1987).[20] A. Howlader and P. Panigrahi, On the distance spectrum of minimal cages and associateddistance biregular graphs, Linear Algebra Appl. 636 (2022), 115–133.[21] A. Kalauch, S. Lavanya, K.C. Sivakumar, Matrices whose group inverses are M-matrices,Linear Algebra Appl. 614 (2021), 44–67.[22] S.J. Kirkland and M. Neumann, Group inverses of M–matrices associated with nonnegativematrices having few eigenvalues, Linear Algebra Appl. 220 (1995), 181–213.[23] S.J. Kirkland and M. Neumann, The M-Matrix Group Generalized Inverse Problem forWeighted Trees, SIAM J. Matrix Anal. Appl., 19 (1998), 226–234.[24] B. Mohar and J. Shawe-Taylor, Distance-biregular graphs with 2-valent vertices and distance-regular line graphs, J. Combin. Theory Ser. B 38(3) (1985), 193–203.[25] H. Urakawa, Heat kernel and Green kernel comparison theorems for infinite graphs, J. Funct.Anal. 146 (1997), 206–235.Universitat Politècnica de Catalunya-BarcelonaTechE-mail address: maria.jose.jimenez@upc.edu

Mean first passage time for distance-biregular graphs

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Mean first passage time for distance-biregular graphs

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