Computer or communication networks are so designed that they do not easily get disrupted under external attack and,moreover,these are easily reconstructible if they do get disrupted.These desirable properties of networks can be measured by various graph parameters,such as connectivity,toughness,scattering number,integrity,tenacity,rupture degree and edge-analogues of some of them.Among these parameters,the tenacity and rupture degree are two better ones to measure the stability of a network.In this paper,we consider two extremal problems on the tenacity of graphs:determine the minimum and maximum tenacity of graphs with given order and size.We give a complete solution to the first problem,while for the second one,it turns out that the problem is much more complicated than that of the minimum case.We determine the maximum tenacity of trees with given order and show the corresponding extremal graphs.The paper concludes with a discussion of a related problem on the edge vulnerability parameters of graphs.
Broersma and Veldman proved that every 2-connected claw-free and P6-free graph is hamil- tonian. Chen et al. extended this result by proving every 2-connected claw-heavy and P6-free graph is hamiltonian. On the other hand, Li et al. constructed a class of 2-connected graphs which are claw-heavy and P6-o-heavy but not hamiltonian. In this paper, we further give some Ore-type degree conditions restricting to induced copies of P6 of a 2-connected claw-heavy graph that can guarantee the graph to be hamiltonian. This improves some previous related results.
In this paper we determine all the bipartite graphs with the maximum sum of squares of degrees among the ones with a given number of vertices and edges.
