Laplacian energy of a graph

Functions of one complex variable II

Preface 1. Introduction 2. Graph operations and modifications 3. Spectrum and structure 4. Characterizations by spectra 5. Structure and one eigenvalue 6. Spectral techniques 7. Laplacians 8. Additional topics 9. Applications Appendix Bibliography Index of symbols Index.

An Introduction to the Theory of Graph Spectra: Spectral techniques

Let T be a set of elements in a ring A. The set T is right permutable if for any a ∈ A and t ∈ T, there exist b ∈ A, u ∈ T such that au = tb. A multiplicative set in a ring A is any subset T of A such that 1 ∈ T,0 ∉ T and T is closed under multiplication. A completely prime ideal in a ring A is any proper ideal B such that A\B is a multiplicative set (i.e. A/Bis a domain). A minimal prime ideal (resp. minimal completely prime ideal) in a ring A is any prime (resp. completely prime) ideal P such that P contains no properly any other prime ideal (resp. completely prime ideal) of A. Let I be any proper ideal of a ring A. The set of all elements a ∈ A such that a + I is a regular element of A/I is denoted by c(I). In particular, c(0) is the set of all regular elements of A.

Rings of quotients

The general Randic index Rα(G) of a (chemical) graph G, is defined as the sum of the weights (d(u)d(v))α of all edges uv of G, where d(u) denotes the degree of a vertex u in G and α an arbitrary real number, which is called the Randic index or connectivity index (or branching index) for α = −1/2 proposed by Milan Randic in 1975. The paper outlines the results known for the (general) Randic index of (chemical) graphs. Some very new results are released. We classify the results into the following categories: extremal values and extremal graphs of Randic index, general Randic index, zeroth-order general Randic index, higher-order Randic index. A few conjectures and open problems are mentioned.

A Survey on the Randic Index

Estimating the Estrada index

(1998). Rings characterized by (pre)envelopes and (pre)covers of their modules. Communications in Algebra: Vol. 26, No. 3, pp. 899-912.

Rings characterized by (pre)envelopes and (pre)covers of their modules

We extend the concept of energy to directed graphs in such a way that Coulson's integral formula remains valid. As a consequence, it is shown that the energy is increasing over the set of digraphs with n vertices and cycles of length h, with respect to a quasi-order relation. Applications to the problem of extremal values of the energy in various classes of digraphs are considered.

Juan Rada

Papers

Estimating the Estrada index

Rings characterized by (pre)envelopes and (pre)covers of their modules

Energy of digraphs

Vertex-degree based topological indices of digraphs

Upper and lower bounds for the energy of bipartite graphs