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A. M. Encinas

Publications and source records attributed to A. M. Encinas.

5 recordsLinked to original sources
arXiv2026

Kemeny's constant via matrix compression and eigenvalue interlacing

Kemeny's constant quantifies the expected time for a random walk to reach a randomly chosen vertex, capturing global properties of a Markov chain. We develop a matrix-analytic framework for bounding Kemeny's constant of a finite connected weighted graph using degree-weighted compressions of the normalized adjacency matrix, pinching inequalities, and eigenvalue interlacing. Our main partition theorem gives lower bounds in terms of the compressed spectrum, with complete equality characterizations, and converts structural graph information into spectrally computable estimates. For instance, when applied to proper color partitions, the proposed method yields a sharp lower bound on Kemeny's constant in terms of the chromatic number \[ K(G)\ge n-2+\frac{1}{χ(G)}, \] which extends a bipartite bound of Ciardo, Dahl, and Kirkland (2022) to arbitrary chromatic number, and which is incomparable with the normalized Hoffman bound by Chung (1997). For unweighted graphs, we also characterize all equality cases of this bound. We further illustrate the power of the proposed matrix framework by deriving spectral bounds for NP-hard graph problems involving normalized cuts and conductance. Our results include an asymptotically sharp conductance bound, two-sided interlacing bounds from principal submatrices and quotient matrices, and estimates for the change in Kemeny's constant under connectivity-preserving deletion of multiple edges.

math.CO↗
arXiv2018

Explicit inverse of nonsingular Jacobi matrices

We present here the necessary and sufficient conditions for the invertibility of tridiagonal matrices, commonly named Jacobi matrices, and explicitly compute their inverse. The techniques we use are related with the solution of Sturm-Liouville boundary value problems associated to second order linear difference equations. These boundary value problems can be expressed throughout a discrete Schrödinger operator and their solutions can be computed using recent advances in the study of linear difference equations. The conditions that ensure the uniqueness solution of the boundary value problem lead us to the invertibility conditions for the matrix, whereas the solutions of the boundary value problems provides the entries of the inverse matrix.

math.RA↗
arXiv2015

The inverse matrix of some circulant matrices

We present here necessary and sufficient conditions for the invertibility of circulant and symmetric matrices that depend on three parameters and moreover, we explicitly compute the inverse. The techniques we use are related with the solution of boundary value problems associated to second order linear difffference equations. Consequently, we reduce the computational cost of the problem. In particular, we recover the inverses of some well known circulant matrices whose coeffifficients are arithmetic or geometric sequences, Horadam numbers among others. We also characterize when a general symmetric circulant and tridiagonal matrix is invertible and in this case, we compute explicitly its inverse.

math.CA↗
arXiv2015

Green Operators of Networks with a new vertex

Any elliptic operator defines an automorphism on the orthogonal subspace to the eigenfunctions associated with the lowest eigenvalue, whose inverse is the orthogonal Green operator. In this study, we show that elliptic Schrödinger operators on networks that have been obtained by adding a new vertex to a given network, can be seen as perturbations of the Schrödinger operators on the initial network. Therefore, the Green function on the new network can be computed in terms of the Green function of the original network.

math.SP↗