Intriguing and tight sets of vertices of point-line geometries have recently been studied in the literature. In this paper, we indicate a more general framework for dealing with these notions. Indeed, we show that some of the results obtained earlier can be generalized to larger classes of graphs. We also give some connections and relations with other notions and results from algebraic graph th...

Association schemes are a central notion in algebraic combinatorics. They provide a unified approach to various questions from design theory, coding theory, algebraic graph theory, discrete geometry, group theory and representation theory. This is facilitated by viewing ideas from the theory of groups and their representations from a combinatorial perspective, thus leading to a more general pic...

In this paper we obtain several tight bounds on different types of alliance numbers of a graph, namely (global) defensive alliance number, global offensive alliance number and global dual alliance number. In particular, we investigate the relationship between the alliance numbers of a graph and its algebraic connectivity, its spectral radius, and its Laplacian spectral radius.

Bidirectional model transformation plays an important role in maintaining consistency between two models, and has many potential applications in software development, including model synchronization, round-trip engineering, software evolution, multiple-view software development, and reverse engineering. However, unclear bidirectional semantics, weak bidirectionalization method, and lack of syst...

We study the behavior of algebraic connectivity in a weighted graph that is subject to site percolation, random deletion of the vertices. Using a refined concentration inequality for random matrices we show in our main theorem that the (augmented) Laplacian of the percolated graph concentrates around its expectation. This concentration bound then provides a lower bound on the algebraic connecti...

We prove that the subdifferential of any semi-algebraic extended-real-valued function on Rn has n-dimensional graph. We discuss consequences for generic semi-algebraic optimization problems.

Alice has p indistinguishable, finitary random oracles, but she knows she can trust only q of them. She wants to simulate a biased coin flip as a function of the oracles’ outputs. We prove: If 1 ≤ q ≤ p/2, the bias can be any rational number and nothing else; if p/2 < q < p, the bias can be any algebraic number and nothing else. The proof uses projective varieties, convex geometry, and the prob...

Let G be a graph on n vertices which has k cutpoints. A tight upper bound on the algebraic connectivity of G in terms of n and k for the case that k > n/2 is provided; the graphs which yield equality in the bound are also characterized. This completes an investigation initiated by the author in a previous paper, which dealt with the corresponding problem for the case that k ≤ n/2.

In this paper, the authors investigate the algebraic connectivity of connected graphs, and determine the graph which has the minimum algebraic connectivity among all connected graphs of order n with given stability number α ≥ d 2 e, or covering number, respectively.