On Distance Energy of Graphs

The D-eigenvalues of a graph G are the eigenvalues of its distance matrix D, and the D-energy ED(G) is the sum of the absolute values of its D-eigenvalues. Two graphs are said to be D-equienergetic if they have the same D-energy. In this note we obtain bounds for the distance spectral radius and D-energy of graphs of diameter 2. Pairs of equiregular D-equienergetic graphs of diameter 2, on p = 3t + 1 vertices are also constructed.

ON DISTANCE ENERGY OF GRAPHS

The D-eigenvalues of a graph G are the eigenvalues of its distance matrix D, and the D-energy ED(G) is the sum of the absolute values of its D-eigenvalues. Two graphs are said to be D-equienergetic if they have the same D-energy. In this note we obtain bounds for the distance spectral radius and D-energy of graphs of diameter 2. Pairs of equiregular D-equienergetic graphs of diameter 2, on p = 3t + 1 vertices are also constructed.

Study on the determination of molecular distance in organic dye mixtures using dual beam thermal lens technique

A sensitive method based on the principle of photothermal phenomena to study the energy transfer processes in organic dye mixtures is presented. A dual beam thermal lens method can be very effectively used as an alternate technique to determine the molecular distance between donor and acceptor in fluorescein–rhodamine B mixture using optical parametric oscillator.

A Beacon based Localization System for Autonomous Mobile Robots and its Applications in Traffic and Transport Control

ACCURATE sensing of vehicle position and attitude is still a very challenging problem in many mobile robot applications. The mobile robot vehicle applications must have some means of estimating where they are and in which direction they are heading. Many existing indoor positioning systems are limited in workspace and robustness because they require clear lines-of-sight or do not provide absolute, driftfree measurements.The research work presented in this dissertation provides a new approach to position and attitude sensing system designed specifically to meet the challenges of operation in a realistic, cluttered indoor environment, such as that of an office building, hospital, industrial or warehouse. This is accomplished by an innovative assembly of infrared LED source that restricts the spreading of the light intensity distribution confined to a sheet of light and is encoded with localization and traffic information. This Digital Infrared Sheet of Light Beacon (DISLiB) developed for mobile robot is a high resolution absolute localization system which is simple, fast, accurate and robust, without much of computational burden or significant processing. Most of the available beacon's performance in corridors and narrow passages are not satisfactory, whereas the performance of DISLiB is very encouraging in such situations. This research overcomes most of the inherent limitations of existing systems.The work further examines the odometric localization errors caused by over count readings of an optical encoder based odometric system in a mobile robot due to wheel-slippage and terrain irregularities. A simple and efficient method is investigated and realized using an FPGA for reducing the errors. The detection and correction is based on redundant encoder measurements. The method suggested relies on the fact that the wheel slippage or terrain irregularities cause more count readings from the encoder than what corresponds to the actual distance travelled by the vehicle.The application of encoded Digital Infrared Sheet of Light Beacon (DISLiB) system can be extended to intelligent control of the public transportation system. The system is capable of receiving traffic status input through a GSM (Global System Mobile) modem. The vehicles have infrared receivers and processors capable of decoding the information, and generating the audio and video messages to assist the driver. The thesis further examines the usefulness of the technique to assist the movement of differently-able (blind) persons in indoor or outdoor premises of his residence.The work addressed in this thesis suggests a new way forward in the development of autonomous robotics and guidance systems. However, this work can be easily extended to many other challenging domains, as well.

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

Equal opportunity networks, distance-balanced graphs, and Wiener game

Given a graph G and a set X ⊆ V(G), the relative Wiener index of X in G is defined as WX (G) = {u,v}∈X 2  dG(u, v) . The graphs G (of even order) in which for every partition V(G) = V1 +V2 of the vertex set V(G) such that |V1| = |V2| we haveWV1 (G) = WV2 (G) are called equal opportunity graphs. In this note we prove that a graph G of even order is an equal opportunity graph if and only if it is a distance-balanced graph. The latter graphs are known by several characteristic properties, for instance, they are precisely the graphs G in which all vertices u ∈ V(G) have the same total distance DG(u) = v∈V(G) dG(u, v). Some related problems are posed along the way, and the so-called Wiener game is introduced.