This paper studies the concepts of uniquely colorable graphs & Perfect graphs. The main results are 1) Every uniquely k-colorable graph is (k 1)-connected. 2) If G is a uniquely k-colorable graph, then  (G) ≥ k l. 3) A maximal planar graph G of order 3 or more has chromatic number 3 if and only if G is Eulerian. 4) Every interval graph is perfect. 5) A graph G is chordal if and only if G can b...

The class of acyclic clique-interval (ACI) graphs is introduced as the class of those graphs G=(V ,E) whose cliques are intervals (chains) of an acyclic order on the vertex set V . The class of ACI graphs is related to the classes of proper interval graphs, tree-clique graphs and to the class DV (intersection graphs of directed paths of a directed tree). Compatibility between a graph and an acy...

We show that if a tolerance graph is the complement of a comparability graph it is a trapezoid graph, i.e., the complement of an order of interval dimension at most 2. As consequences we are able to give obstructions for the class of bounded tolerance graphs and to give an example of a graph which is alternatingly orientable but not a tolerance graph. We also characterize the tolerance graphs a...

We study random k-lifts of large, but otherwise arbitrary graphs G. We prove that, with high probability, all eigenvalues of the adjacency matrix of the lift that are not eigenvalues of G are of the order of √ ΔG ln(kn), where ΔG is the maximum degree of G. Similarly, and also with high probability, the “new” eigenvalues of the normalized Laplacian of G are all in an interval of length O( √ ln(...

The average distance of a vertex $v$ of a connected graph $G$is the arithmetic mean of the distances from $v$ to allother vertices of $G$. The proximity $pi(G)$ and the remoteness $rho(G)$of $G$ are defined as the minimum and maximum averagedistance of the vertices of $G$. In this paper we investigate the difference between proximity or remoteness and the classical distanceparameters diameter a...