Diluted three-dimensional random field Ising model at zero temperature with metastable dynamics.

The influence of vacancy concentration on the behavior of the three-dimensional random field Ising model with metastable dynamics is studied. We have focused our analysis on the number of spanning avalanches which allows us a clear determination of the critical line where the hysteresis loops change from continuous to discontinuous. By a detailed finite-size scaling analysis we determine the phase diagram and numerically estimate the critical exponents along the whole critical line. Finally, we discuss the origin of the curvature of the critical line at high vacancy concentration.

Influence of the driving mechanism on the response of systems with athermal dynamics: The example of the random-field Ising model

We investigate the influence of the driving mechanism on the hysteretic response of systems with athermal dynamics. In the framework of local mean-field theory at finite temperature (but neglecting thermally activated processes), we compare the rate-independent hysteresis loops obtained in the random field Ising model when controlling either the external magnetic field H or the extensive magnetization M. Two distinct behaviors are observed, depending on disorder strength. At large disorder, the H-driven and M-driven protocols yield identical hysteresis loops in the thermodynamic limit. At low disorder, when the H-driven magnetization curve is discontinuous (due to the presence of a macroscopic avalanche), the M-driven loop is reentrant while the induced field exhibits strong intermittent fluctuations and is only weakly self-averaging. The relevance of these results to the experimental observations in ferromagnetic materials, shape memory alloys, and other disordered systems is discussed.

On infinite graphs and related matrices

This thesis Entitled On Infinite graphs and related matrices.ln the last two decades (iraph theory has captured wide attraction as a Mathematical model for any system involving a binary relation. The theory is intimately related to many other branches of Mathematics including Matrix Theory Group theory. Probability. Topology and Combinatorics . and has applications in many other disciplines..Any sort of study on infinite graphs naturally involves an attempt to extend the well known results on the much familiar finite graphs. A graph is completely determined by either its adjacencies or its incidences. A matrix can convey this information completely. This makes a proper labelling of the vertices. edges and any other elements considered, an inevitable process. Many types of labelling of finite graphs as Cordial labelling, Egyptian labelling, Arithmetic labeling and Magical labelling are available in the literature. The number of matrices associated with a finite graph are too many For a study ofthis type to be exhaustive. A large number of theorems have been established by various authors for finite matrices. The extension of these results to infinite matrices associated with infinite graphs is neither obvious nor always possible due to convergence problems. In this thesis our attempt is to obtain theorems of a similar nature on infinite graphs and infinite matrices. We consider the three most commonly used matrices or operators, namely, the adjacency matrix

Random-field Potts model with dipolarlike interactions: Hysteresis avalanches and microstructure

A model for the study of hysteresis and avalanches in a first-order phase transition from a single variant phase to a multivariant phase is presented. The model is based on a modification of the random-field Potts model with metastable dynamics by adding a dipolar interaction term truncated at nearest neighbors. We focus our study on hysteresis loop properties, on the three-dimensional microstructure formation, and on avalanche statistics.

‘Studies on some topics in product graphs’

For routing problems in interconnection networks it is important to find the shortest containers between any two vertices, since the w-wide diameter gives the maximum communication delay when there are up to w−1 faulty nodes in a network modeled by a graph. The concept of ‘wide diameter’ was introduced by Hsu [41] to unify the concepts of diameter and The concept of ‘domination’ has attracted interest due to its wide applications in many real world situations [38]. A connected dominating set serves as a virtual backbone of a network and it is a set of vertices that helps in routing. In this thesis, we make an earnest attempt to study some of these notions in graph products. This include, the diameter variability, the diameter vulnerability, the component factors and the domination criticality.connectivity

Computing median and antimedian sets in median graphs

The median (antimedian) set of a profile π = (u1, . . . , uk) of vertices of a graphG is the set of vertices x that minimize (maximize) the remoteness i d(x,ui ). Two algorithms for median graphs G of complexity O(nidim(G)) are designed, where n is the order and idim(G) the isometric dimension of G. The first algorithm computes median sets of profiles and will be in practice often faster than the other algorithm which in addition computes antimedian sets and remoteness functions and works in all partial cubes

On the remoteness function in median graphs

A profile on a graph G is any nonempty multiset whose elements are vertices from G. The corresponding remoteness function associates to each vertex x 2 V.G/ the sum of distances from x to the vertices in the profile. Starting from some nice and useful properties of the remoteness function in hypercubes, the remoteness function is studied in arbitrary median graphs with respect to their isometric embeddings in hypercubes. In particular, a relation between the vertices in a median graph G whose remoteness function is maximum (antimedian set of G) with the antimedian set of the host hypercube is found. While for odd profiles the antimedian set is an independent set that lies in the strict boundary of a median graph, there exist median graphs in which special even profiles yield a constant remoteness function. We characterize such median graphs in two ways: as the graphs whose periphery transversal number is 2, and as the graphs with the geodetic number equal to 2. Finally, we present an algorithm that, given a graph G on n vertices and m edges, decides in O.mlog n/ time whether G is a median graph with geodetic number 2

Strongly distance-balanced graphs and graph products

A graph G is strongly distance-balanced if for every edge uv of G and every i 0 the number of vertices x with d.x; u/ D d.x; v/ 1 D i equals the number of vertices y with d.y; v/ D d.y; u/ 1 D i. It is proved that the strong product of graphs is strongly distance-balanced if and only if both factors are strongly distance-balanced. It is also proved that connected components of the direct product of two bipartite graphs are strongly distancebalanced if and only if both factors are strongly distance-balanced. Additionally, a new characterization of distance-balanced graphs and an algorithm of time complexity O.mn/ for their recognition, wheremis the number of edges and n the number of vertices of the graph in question, are given

Simultaneous Embeddings Of Graphs As Median And Antimedian Subgraphs

The distance DG(v) of a vertex v in an undirected graph G is the sum of the distances between v and all other vertices of G. The set of vertices in G with maximum (minimum) distance is the antimedian (median) set of a graph G. It is proved that for arbitrary graphs G and J and a positive integer r 2, there exists a connected graph H such that G is the antimedian and J the median subgraphs of H, respectively, and that dH(G, J) = r. When both G and J are connected, G and J can in addition be made convex subgraphs of H.

The Plurality Strategy on Graphs

The Majority Strategy for finding medians of a set of clients on a graph can be relaxed in the following way: if we are at v, then we move to a neighbor w if there are at least as many clients closer to w than to v (thus ignoring the clients at equal distance from v and w). The graphs on which this Plurality Strategy always finds the set of all medians are precisely those for which the set of medians induces always a connected subgraph

Almost Self-Centered Median And Chordal Graphs

Almost self-centered graphs were recently introduced as the graphs with exactly two non-central vertices. In this paper we characterize almost selfcentered graphs among median graphs and among chordal graphs. In the first case P4 and the graphs obtained from hypercubes by attaching to them a single leaf are the only such graphs. Among chordal graph the variety of almost self-centered graph is much richer, despite the fact that their diameter is at most 3. We also discuss almost self-centered graphs among partial cubes and among k-chordal graphs, classes of graphs that generalize median and chordal graphs, respectively. Characterizations of almost self-centered graphs among these two classes seem elusive

The median function on graphs with bounded profiles

The median of a profile = (u1, . . . , uk ) of vertices of a graph G is the set of vertices x that minimize the sum of distances from x to the vertices of . It is shown that for profiles with diameter the median set can be computed within an isometric subgraph of G that contains a vertex x of and the r -ball around x, where r > 2 − 1 − 2 /| |. The median index of a graph and r -joins of graphs are introduced and it is shown that r -joins preserve the property of having a large median index. Consensus strategies are also briefly discussed on a graph with bounded profiles.