Forest matrices around the Laplacian matrix

We study the matrices Q_k of in-forests of a weighted digraph G and their connections with the Laplacian matrix L of G. The (i,j) entry of Q_k is the total weight of spanning converging forests (in-forests) with k arcs such that i belongs to a tree rooted at j. The forest matrices, Q_k, can be calculated recursively and expressed by polynomials in the Laplacian matrix; they provide representations for the generalized inverses, the powers, and some eigenvectors of L. The normalized in-forest matrices are row stochastic; the normalized matrix of maximum in-forests is the eigenprojection of the Laplacian matrix, which provides an immediate proof of the Markov chain tree theorem. A source of these results is the fact that matrices Q_k are the matrix coefficients in the polynomial expansion of adj(a*I+L). Thereby they are precisely Faddeev's matrices for -L. Keywords: Weighted digraph; Laplacian matrix; Spanning forest; Matrix-forest theorem; Leverrier-Faddeev method; Markov chain tree theorem; Eigenprojection; Generalized inverse; Singular M-matrixComment: 19 pages, presented at the Edinburgh (2001) Conference on Algebraic Graph Theor

ZAGREB INDICES OF A NEW SUM OF GRAPHS

The first and second Zagreb indices, since its inception have been subjected to an extensive research in the physio- chemical analysis of compounds. In [6] Hanyuan Deng et.al computed the first and second Zagreb indices of four new operations on a graph defined by M. Eliasi, B. Taeri in [4]. Motivated from this we define a new operation on graphs and compute the first and second Zagreb indices of the resultant graph. We illustrate the results with some examples

On the new extension of distance-balanced graphs

‎In this paper‎, ‎we initially introduce the concept of $n$-distance-balanced property which is considered as the generalized concept of distance-balanced property‎. ‎In our consideration‎, ‎we also define the new concept locally regularity in order to find a connection between $n$-distance-balanced graphs and their lexicographic product‎. ‎Furthermore‎, ‎we include a characteristic method which is practicable and can be used to classify all graphs with $i$-distance-balanced properties for $i=2,3$ which is also relevant to the concept of total distance‎. ‎Moreover‎, ‎we conclude a connection between distance-balanced and 2-distance-balanced graphs‎

Bounds on distance-based topological indices in graphs.

Thesis (Ph.D.)-University of KwaZulu-Natal, Westville, 2012.This thesis details the results of investigations into bounds on some distance-based topological indices. The thesis consists of six chapters. In the first chapter we define the standard graph theory concepts, and introduce the distance-based graph invariants called topological indices. We give some background to these mathematical models, and show their applications, which are largely in chemistry and pharmacology. To complete the chapter we present some known results which will be relevant to the work. Chapter 2 focuses on the topological index called the eccentric connectivity index. We obtain an exact lower bound on this index, in terms of order, and show that this bound is sharp. An asymptotically sharp upper bound is also derived. In addition, for trees of given order, when the diameter is also prescribed, tight upper and lower bounds are provided. Our investigation into the eccentric connectivity index continues in Chapter 3. We generalize a result on trees from the previous chapter, proving that the known tight lower bound on the index for a tree in terms of order and diameter, is also valid for a graph of given order and diameter. In Chapter 4, we turn to bounds on the eccentric connectivity index in terms of order and minimum degree. We first consider graphs with constant degree (regular graphs). Došlić, Saheli & Vukičević, and Ilić posed the problem of determining extremal graphs with respect to our index, for regular (and more specifically, cubic) graphs. In addressing this open problem, we find upper and lower bounds for the index. We also provide an extremal graph for the upper bound. Thereafter, the chapter continues with a consideration of minimum degree. For given order and minimum degree, an asymptotically sharp upper bound on the index is derived. In Chapter 5, we turn our focus to the well-studied Wiener index. For trees of given order, we determine a sharp upper bound on this index, in terms of the eccentric connectivity index. With the use of spanning trees, this bound is then generalized to graphs. Yet another distance-based topological index, the degree distance, is considered in Chapter 6. We find an asymptotically sharp upper bound on this index, for a graph of given order. This proof definitively settles a conjecture posed by Tomescu in 1999

Mining and analysis of real-world graphs

Networked systems are everywhere - such as the Internet, social networks, biological networks, transportation networks, power grid networks, etc. They can be very large yet enormously complex. They can contain a lot of information, either open and transparent or under the cover and coded. Such real-world systems can be modeled using graphs and be mined and analyzed through the lens of network analysis. Network analysis can be applied in recognition of frequent patterns among the connected components in a large graph, such as social networks, where visual analysis is almost impossible. Frequent patterns illuminate statistically important subgraphs that are usually small enough to analyze visually. Graph mining has different practical applications in fraud detection, outliers detection, chemical molecules, etc., based on the necessity of extracting and understanding the information yielded. Network analysis can also be used to quantitatively evaluate and improve the resilience of infrastructure networks such as the Internet or power grids. Infrastructure networks directly affect the quality of people\u27s lives. However, a disastrous incident in these networks may lead to a cascading breakdown of the whole network and serious economic consequences. In essence, network analysis can help us gain actionable insights and make better data-driven decisions based on the networks. On that note, the objective of this dissertation is to improve upon existing tools for more accurate mining and analysis of real-world networks --Abstract, page iv