http://nova.newcastle.edu.au/vital/access/services/Feed ${session.getAttribute("locale")} 5 http://nova.newcastle.edu.au/vital/access/manager/Repository/uon:9241 By the extremal numberex(v;{C₃,C₄,…,Cn}) we denote the maximum number of edges in a graph of order v and girth at least g≥n+1. The set of such graphs is denoted by . In 1975, Erdős mentioned the problem of determining extremal numbers ex(v;{C₃,C₄}) in a graph of order v and girth at least five. In this paper, we consider a generalized version of the problem for any value of girth by using the hybrid simulated annealing and genetic algorithm (HSAGA). Using this algorithm, some new results for n≥5 have been obtained. In particular, we generate some graphs of girth 6,7 and 8 which in some cases have more edges than corresponding cages. Furthermore, future work will be described regarding the investigation of structural properties of such extremal graphs and the implementation of HSAGA using parallel computing. 2012-01-30T05:00:23.221Z ]]> http://nova.newcastle.edu.au/vital/access/manager/Repository/uon:5502 A (k,g)-cage is a k-regular graph with girth g and with the least possible number of vertices. In this paper we give a brief overview of the current results on the connectivity of (k,g)-cages and we improve the current known best lower bound on the vertex connectivity of (k,g)-cages for g even. 2012-01-30T04:04:28.862Z ]]> http://nova.newcastle.edu.au/vital/access/manager/Repository/uon:7829 A (k,g)-cage is a k-regular graph of girth g and with the least possible number of vertices. In this paper, we investigate the problem of how many connected components there will be after removing a cutset of up to k vertices from a (k,g)-cage. 2011-11-02T03:30:07.635Z ]]> http://nova.newcastle.edu.au/vital/access/manager/Repository/uon:8038 A graph is superconnected, for short super-κ, if all minimum vertex-cuts consist of the vertices adjacent with one vertex. In this paper we prove for any r-regular graph of diameter D and odd girth g that if D≤g−2, then the graph is super-κ when g≥5 and a complete graph otherwise. 2011-07-04T06:20:12.240Z ]]> http://nova.newcastle.edu.au/vital/access/manager/Repository/uon:379 A ([delta],g)-cage is a [delta]-regular graph with girth g and with the least possible number of vertices. We prove that all ([delta],g)-cages are r-connected with for g[greater-or-equal, slanted]7 odd. This result supports the conjecture of Fu, Huang and Rodger that all ([delta];g)-cages are [delta]-connected. 2010-04-27T05:46:32.936Z ]]> http://nova.newcastle.edu.au/vital/access/manager/Repository/uon:4799 A vertex-cut X is said to be a restricted cut of a graph G if it is a vertex-cut such that no vertex u in G has all its neighbors in X. Clearly, each connected component of G−X must have at least two vertices. The restricted connectivity κ'(G) of a connected graph G is defined as the minimum cardinality of a restricted cut. Additionally, if the deletion of a minimum restricted cut isolates one edge, then the graph is said to be super-restricted connected. In this paper, several sufficient conditions yielding super-restricted connected graphs are given in terms of the girth and the diameter. The corresponding problem for super-edge-restricted-connected graph is also studied. 2010-04-27T05:33:13.998Z ]]>