AUTOMATED CONJECTURING ON THE INDEPENDENCE NUMBER AND MINIMUM DEGREE OF DIAMETER-2-CRITICAL GRAPHS

A diameter-2-critical (D2C) graph is a graph with diameter two such that removing any edge increases the diameter or disconnects the graph. In this paper, we look at other lesser-studied properties of D2C graphs, focusing mainly on their independence number and minimum degree. We show that there exist D2C graphs with minimum degree strictly larger than their independence number, and that this gap can be arbitrarily large. We also exhibit D2C graphs with maximum number of common neighbors strictly greater than their independence number, and that this gap can be arbitrarily large. Furthermore, we exhibit a D2C graph whose number of distinct degrees in its degree sequence is strictly greater than its independence number. Additionally, we characterize D2C graphs with independence number 2 and show that all such graphs have independence number greater or equal to their minimum degree

The connectivity of a graph and its complement

AbstractLet G be a graph with minimum degree δ(G), edge-connectivity λ(G), vertex-connectivity κ(G), and let Ḡ be the complement of G.In this article we prove that either λ(G)=δ(G) or λ(Ḡ)=δ(Ḡ). In addition, we present the Nordhaus–Gaddum type result κ(G)+κ(Ḡ)≥min{δ(G),δ(Ḡ)}+1. A family of examples will show that this inequality is best possible