# José Luis PalaciosUniversity of New Mexico | UNM · Department of Electrical and Computer Engineering

José Luis Palacios

PhD

## About

76

Publications

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Introduction

**Skills and Expertise**

## Publications

Publications (76)

We find new upper bounds for the global cyclicity index, a variant of the Kirchhoff index, and discuss the wide family of graphs for which the bounds are attained.

We propose a new molecular index based on a distance on graphs defined through hitting times of random walks on connected graphs. We show the conections to previous probabilistic/electric indices such as the RW inex and the Kirchhoff index, compute its value for some families of graphs, and present some open questions.

We find closed-form expressions for the variance and the third moment of the number of hires in the assistant hiring algorithm, as well as asymptotic values for higher moments of this variable.

We find closed form formulas for the Kemeny's constant and the Kirchhoff index for the cluster $G_1\{G_2\}$ of two highly symmetric graphs $G_1$, $G_2$, in terms of the parameters of the original graphs. We also discuss some necessary conditions for a graph to be highly symmetric.

We find closed form formulas for Kemeny's constant and its relationship with two Kirchhoffian indices for some composite graphs that use as basic building block a graph endowed with one of several symmetry properties.

In the famous paper (Brooks et al., 1940) an equivalence was established between planar electric networks and tilings of cylinders. However, the proof given there is rather difficult and requires knowledge of a larger theory. We give a simple new proof of this equivalence. Furthermore, there is a well known relationship between electric networks an...

We provide upper bounds for the atom-bond connectivity index that correct some bounds given in [6], using the same majorization technique.

Using majorization, we find lower bounds for the geometric–arithmetic index of graphs containing either pendant or fully connected vertices.

The resistance indices, namely the Kirchhoff index and its generalisations, have undergone intense critical scrutiny in recent years. Based on random walks, we derive three Kirchhoffian indices for strongly connected and weighted digraphs. These indices are expressed in terms of (i) hitting times and (ii) the trace and eigenvalues of suitable matri...

Given a simple undirected graph G=(V,E) with n vertices, if for the largest eigenvalue of its Laplacian matrix λ1 there exists a lower bound λ1≥α≥dGnn−1, then we have that its Laplacian energy satisfies LE(G)≥max{2dG,2(α−dG)},
where dG=d1+⋯dnn is the average degree of G. This generic lower bound, obtained with the majorization technique, allows us...

Using majorization we find two general lower bounds for the Laplacian Resolvent Energy of a graph, one in terms of the degrees of the vertices, the other in terms of the number of edges, and some particular lower bounds for c-cyclic graphs, 0 ≤ c ≤ 6.

Using majorization, we give new bounds for the resolvent energy of general bipartite graphs. We also find more specific lower bounds in the particular case of trees. © 2018 University of Kragujevac, Faculty of Science. All rights reserved.

The n-tuple of Laplacian eigenvalues of a graph majorizes the n-tuple of its degrees. This simple fact allows us to obtain a set of inequalities - some known, some new - for several descriptors given in terms of the Laplacian eigenvectors, in a unified manner. © 2017, University of Kragujevac, Faculty of Science. All rights reserved.

For any simple connected undirected graph, and the random walk on it, we obtain a formula for the sum of all expected hitting times – normalized by the stationary distribution – expressed in terms of the eigenvalues of a certain modified Laplacian matrix. This allows us to find lower bounds for these sums of hitting times, as well as new lower boun...

We revise some bounds found in [2] and give a new general upper bound for the multiplicative degree-Kirchhoff index.

For a connected undirected graph G = (V, E) with vertex set {1, 2, ..., n} and degrees d(i), for 1 <= i <= n, we show that ABC(G) <= root(n - 1)(vertical bar E vertical bar - R-1(G)), where R-1(G) = Sigma((i,j)is an element of E) 1/d(i)d(j) is the Randic index. This bound allows us to obtain some maximal results for the ABC index with elementary pr...

We introduce the mixed degree-Kirchhoff index, a new molecular descriptor defined by (Formula presented) where di is the degree of the vertex i and Rij is the effective resistance between vertices i and j. We give general upper and lower bounds for Ȓ(G) and show that, unlike other related descriptors, it attains its largest asymptotic value (order...

Using the electric approach, we derive a formula that expresses an expected hitting time of a random walk between two vertices a and b of a graph G as a convex linear combination of expected hitting times of walks between a and b on subgraphs of G, provided certain condition on a and b is satisfied. Then we use this formula in several applications.

We identify the graphs of diameter 2 with largest, second largest, third largest, smallest, second smallest and third smallest Kirchhoff indices. The main tools are standard results of electric networks.

Every birth and death chain on a finite tree can be represented as a random walk on the underlying tree endowed with appropriate conductances. We provide an algorithm that finds these conductances in linear time. Then, using the electric network approach, we find the values for the stationary distribution and for the expected hitting times between...

We derive formulas for the expected hitting times of general random walks on graphs, in terms of voltages, with very elementary electric means. Under this new light we revise bounds and hitting times for birth-and-death Markov chains and for walks on graphs with cutpoints, and give some exact computations on the necklace graph. We also prove Tetali...

Let G be a connected undirected graph with vertex set {1, 2,.., n} and degrees di, for 1 ≤ i ≤ n. Then we show that where Rij is the effective resistance between i and j. This general bound allows us to obtain many other particular bounds and asymptotic maximal results for the ABC index with elementary proofs.

Making use of a majorization technique for a suitable class of graphs, we
derive upper and lower bounds for some topological indices depending on the
degree sequence over all vertices, namely the first general Zagreb index and
the first multiplicative Zagreb index. Specifically, after characterizing
$c-$cyclic graphs $(0\leq c\leq 6)$ as those whos...

Given a simple connected graph on $N$ vertices with size $|E|$ and degree
sequence $d_{1}\leq d_{2}\leq ...\leq d_{N}$, the aim of this paper is to
exhibit new upper and lower bounds for the additive degree-Kirchhoff index in
closed forms, not containing effective resistances but a few invariants
$(N,|E|$ and the degrees $d_{i}$) and applicable in...

Using a majorization technique that identifies the maximal and minimal vectors of a variety of subsets of \({\mathbb{R}^{n}}\) , we find upper and lower bounds for the Kirchhoff index K(G) of an arbitrary simple connected graph G that improve those existing in the literature. Specifically we show that $$K(G) \geq \frac{n}{d_{1}} \left[ \frac{1}{1+\...

We give tight upper and lower bounds for the additive degree-Kirchhoff index of a connected undirected graph.

Given a simple connected graph G, this paper presents a new approach for localizing the graph topological indices given by the sum of the α-th power of the non zero normalized Laplacian eigenvalues. Through this method, old and new bounds are derived, showing how former results in the literature can be improved.

We conjecture that if TjTj is the hitting time of vertex jj then ∑jEiTj≥(N−1)2, for all ii, for a random walk on any connected graph G=(V,E)G=(V,E) with |E|=N|E|=N. We prove the conjecture for a family of graphs containing the regular graphs and obtain slightly better bounds for trees and non-regular edge-transitive graphs.

We derive a closed-form formula for the expected hitting times of general random walks on trees with very elementary electric means, and discuss when these expected hitting times turn out to be natural numbers in the case of the linear graph endowed with conductances which are all natural numbers.

We use a simple three-state Markov chain model in order to study the number of runs or spells of hot, cold and neutral months for El Niño/Southern Oscillation (ENSO) and for the North Atlantic Oscillation (NAO), as reported in the page http://www.cdc.noaa.gov/ClimateIndices/List/. We discuss the fit of the Markovian model to both phenomena and the...

Simple random walks probabilistically grown step by step on a graph are distinguished from walk enumerations and associated equipoise random walks. Substructure characteristics and graph invariants correspondingly defined for the two types of random walks are then also distinct, though there often are analogous relations. It is noted that the conne...

We give sufficient conditions under whichfor a sequence of nonnegative random variables and [alpha]>0.

We find closed-form expressions for the resistance, or Kirchhoff index, of certain connected graphs using Foster's theorems, random walks, and the superposition principle. © 2001 John Wiley & Sons, Inc. Int J Quant Chem 81: 135–140, 2001

Using the electric and coupling approaches, we derive a series of results concerning the mixing times for the stratified random walk on the d-cube, inspired in the results of Chung and Graham (1997) Random Structures and Algorithms, 11. Key Words: effective resistance, coupling, birth and death chains 1991 Mathematics Subject Classification. Primar...

Using the electric network approach, we give closed-form formulas for the stationary probabilities and expected hitting times in balanced circular Markov chains. As an application, we give a closed-form formula for the duration of play in the general ruin problem, where the probabilities of winning a particular game depend on the amount of the curr...

We present estimators of extinction and migration rates based on the runs of presences and absences of a species on an island, that behave as well as those based on the 2-state (presence and absence) Markov chain, both regarding their computational and memory complexities, and their asymptotic statistical behavior (we prove that they are strongly c...

We give formulas, in terms of the number of pure k-cycles, for the expected hitting times between vertices at distances greater than 1 for random walks on edge-transitive graphs, extending our prior results for neighboring vertices and also extending results of Devroye-Sbihi and Biggs concerning distance-regular graphs. We apply these formulas to a...

We study the symmetry properties in weak products of graphs which are inherited from the coordinate graphs and which enable the computation of expected hitting times for a random walk on the product graph. We obtain explicit values for expected hitting times between non-neighboring vertices of the product of certain unitary Cayley graphs, showing a...

Using the electric network approach, we give a closed-form formula for the solution of the ruin problem in the case that the probabilities of winning a particular game depend on the amount of the current fortune.

Using the electric network approach, we give a closed-form formula for the solution of the ruin problem in the case that the probabilities of winning a particular game depend on the amount of the current fortune.

We find explicit values for the expected hitting times between neighboring vertices of random walks on edge-transitive graphs, extending prior results and allowing the computation of sharp upper and lower bounds for the expected cover times of those graphs.

Most earlier mathematical studies of baseball required particular models for advancing runners based on a small set of offensive possibilities. Other efforts considered only teams with players of identical ability. We introduce a Markov chain method that considers teams made up of players with different abilities and which is not restricted to a gi...

We show how to apply a pointwise limit theorem in the study of storage measures of search trees, yielding pointwise asymptotics as an alternative to average asymptotics derived through fringe analysis. We show in particular, how for the pointwise versions of the storage measures previously studied in the literature, m-ary trees are outperformed by...

Kullback’s information is shown to be essential in comparing (asymptotically) the entropy of uniform spacings with the entropy of spacings determined by an i.i.d. sequence in [0,1]. A characterization of the uniform distribution based on entropy of sample spacings is obtained and discussed in the context of parametric estimation and Kullback’s prin...

Using the electric network approach, we give closed form formulas for the expected hitting times in (finite and infinite) birth and death chains.

Using the electric network approach, we give closed-form formulas for the expected hitting times in the Ehrenfest urn model.

Using the electric network approach, we give closed-form formulas for the expected hitting times in the Ehrenfest urn model.

Using the electric network approach, we give very simple derivations for the expected first passage from the origin to the opposite vertex in the d-cube (i.e. the Ehrenfest urn model) and the Platonic graphs.

Using the electric network approach, we give very simple derivations for the expected first passage from the origin to the opposite vertex in the d -cube (i.e. the Ehrenfest urn model) and the Platonic graphs.

We give bounds for the covering time of a random walk on an undirected connected graph in terms of the diameter of the graph. The bounds are tight in many instances, particularly when the graph is a tree.

We show how the distribution of the number of heads (mod k) in a sequence of tosses of a coin can be found by looking at the sequence of tosses as a Markov chain, providing as corollaries some non-trivial combinatorial identities.

It is known that for a random walk on a connected graph G on N vertices {xl,…,xN} satisfying υ(xi)≤ k for all i [υ(xi) is the valence of xi] , the maximum expected number of steps to get from one vertex to another has a bound of order kN(N−1). We give simple sufficient conditions under which, even though υ(xi= O>(N) for all i, the expected hitting...

M. M. Ali [J. R. Stat. Soc., Ser. B 42, 162-164 (1980; Zbl 0443.62035)] showed that if (X 1 ,···,X n ) is a random vector with spherically symmetric distribution, independence, of the sample mean and the sample variance is equivalent to independence of the coordinates: this result contains, inter alia, Maxwell’s theorem. It is shown here that Ali’s...

Conditions on the distribution of a process $\{X_n, n \in I \}$ are given under which the invariant, tail and exchangeable σ-fields coincide; the index set I is either the positive integers or all the integers. The results proven here correct similar statements given in [3].

Conditions on the distribution of a process { X n , n ∈ I } \{ {X_n},n \in I\} are given under which the invariant, tail and exchangeable σ \sigma -fields coincide; the index set I I is either the positive integers or all the integers. The results proven here correct similar statements given in [3].