A connected graph G is said to be k-connected if it has more than k vertices and remains whenever fewer are deleted. In this paper, for a with sufficiently large order, we present tight sufficient condition fixed minimum degree based on the Q-index. Our result can viewed as spectral counterpart of corresponding Dirac-type condition.

In the algorithm for the disjoint paths problem given in Graph Minors XIII, we used without proof a lemma that, in solving such a problem, a vertex which was sufficiently “insulated” from the rest of the graph by a large planar piece of the graph was irrelevant, and could be deleted without changing the problem. In this paper we prove the lemma.

A graph G of order n is k-factor-critical, where k is an integer of the same parity as n with 0 ::; k ::; n, if G X has a perfect matching for any set X of k vertices of G. A k-factor-critical graph G is called minimal if for any edge e E E(G), G e is not k-factor-critical. In this paper we study some properties of minimally k-factor-critical graphs, in particular a bound on the minimum degree,...

We prove that a graph admits a strongly 2-connected orientation if and only if it is 4-edge-connected, and every vertex-deleted subgraph is 2-edge-connected. In particular, every 4-connected graph has such an orientation while no cubic 3-connected graph has such an orientation.

A graph G is hypohamiltonian/hypotraceable if it is not hamiltonian/traceable, but all vertex deleted subgraphs of G are hamiltonian/traceable. Until now all hypotraceable graphs were constructed using hypohamiltonian graphs; extending a method of Thomassen [8] we present a construction that uses so-called almost hypohamiltonian graphs (nonhamiltonian graphs, whose vertex deleted subgraphs are ...

If a graph is connected then the largest eigenvalue (i.e., index) generally changes (decreases or increases) if some local modifications are performed. In this paper two types of modifications are considered: (i) for a fixed vertex, t edges incident with it are deleted, while s new edges incident with it are inserted; (ii) for two non-adjacent vertices, t edges incident with one vertex are dele...

We propose a new random graph model—Edge Popularity—for the web graph and other complex networks, where edges are deleted over time and an edge is chosen to be deleted with probability inversely proportional to the in-degree of the destination. We show that with probability tending to one as time tends to infinity, the model generates graphs whose degree distribution follows a power law. Depend...

We propose new evolutionary stochastic models for the web graph and other massive networks, where edges are deleted over time and an edge is chosen to be deleted with probability inversely proportional to the in-degree of the destination. The degree distributions of graphs generated by our models follow a power law. A rigorous proof of power law degree distributions is given using martingales a...

We prove that the class of nontrivial connected strong product graphs is weakly reconstructible. We also show that any nontrivial connected thin strong product graph can be uniquely reconstructed from each of its one-vertex-deleted deleted subgraphs. © 2006 Elsevier B.V. All rights reserved. MSC: 05C

Configuration graphs with node degrees being independent identically distributed random variables following the power-law distribution a shift are studied. We consider conditional that consist of only one connected component. Computer simulations used to study how connectivity such changes under two types destruction processes: “target attack” and “random breakdown”. propose models for dependen...