On the Randi´ c index and conditional parameters of a graph J. A. Rodr´ ıguez ∗ and J. M. Sigarreta † Department of Mathematics Carlos III of Madrid University Avda. de la Universidad 30, 28911 Legan´ es (Madrid), Spain (Received November 26, 2004) Abstract The aim of this paper is to study some parameters of simple graphs related with the degree of the vertices. So, our main tool is the n × n matrix A whose (i, j )-entry is a ij =  1 √ δ i δ j if v i ∼ v j ; 0 otherwise, where δ i denotes the degree of the vertex v i . We study the Randi´ c index and some interesting particular cases of conditional excess, conditional Wiener index, and conditional diameter. In particular, using the matrix A or its eigenvalues, we obtain tight bounds on the studied parameters. 1 Introduction In order to deduce properties of graphs from results and methods of algebra, firstly we need to translate properties of graphs into algebraic properties. In this sense, a natural way is to consider algebraic structures or algebraic objects as, for instance, groups or matrices. In particular, the use of matrices allows us to use methods of linear algebra to derive properties of graphs. There are various matrices that are naturally associated with graphs, such as the adjacency matrix, the Laplacian matrix, and the incidence matrix [1, 3, 8]. One of the main aims of algebraic graph theory is to determine how, or whether, properties of graphs are reflected in the algebraic properties of such matrices [8]. The aim of this paper is to study the Randi´ c index and some interesting particular cases of conditional excess, conditional Wiener index, and conditional diameter. All these parameters are related with the degree of the vertices of the graph. So, our main tool will be a suitable adjacency matrix that we call degree-adjacency matrix. * e-mail:juanalberto.rodriguez@uc3m.es † e-mail:josemaria.sigarreta@uc3m.es 1