Algebraic connectivity for vertex-deleted subgraphs, and a notion of vertex centrality

Algebraic connectivity for vertex-deleted subgraphs, and a notion of vertex centrality

Let G be a connected graph, suppose that v is a vertex of G, and denote the subgraph formed from G by deleting vertex v by G \ v. Denote the algebraic connectivities of G and G \ v by α(G) and α(G \ v), respectively. In this paper, we consider the functions φ(v) = α(G) − α(G \ v) and κ(v) = α(G\v) α(G) , provide attainable upper and lower bounds on both functions, and characterise the equality ...

Vertex-deleted and Edge-deleted Subgraphs

The foremost problem in this area of graph theory is the Reconstruction Conjecture which states that a graph is reconstructible from its collection of vertex-deleted subgraphs. In spite of several attempts to prove the conjecture only very partial results have been obtained. Several of these results do however bring to light interesting structural relationships between a graph and its subgraphs...

Undirected Vertex-Connectivity Structure and Smallest Four-Vertex-Connectivity Augmentation

In this paper we study properties for the structure of an undirected graph that is not vertex connected We also study the evolution of this structure when an edge is added to optimally increase the vertex connectivity of the underlying graph Several properties reported here can be extended to the case of a graph that is not k vertex connected for an arbitrary k Using properties obtained here we...

On the multiplicities of eigenvalues of graphs and their vertex deleted subgraphs: old and new results

Vertex colorings without rainbow subgraphs

Given a coloring of the vertices of a graph G, we say a subgraph is rainbow if its vertices receive distinct colors. For graph F , we define the F -upper chromatic number of G as the maximum number of colors that can be used to color the vertices of G such that there is no rainbow copy of F . We present some results on this parameter for certain graph classes. The focus is on the case that F is...