many graph theoretical parameters have been used to describe the vulnerability of communication networks, including toughness, binding number, rate of disruption, neighbor-connectivity, integrity, mean integrity, edgeconnectivity vector, l-connectivity and tenacity. in this paper we discuss integrity and its properties in vulnerability calculation. the integrity of a graph g, i(g), is defined t...

in this paper, a new molecular-structure descriptor, the general sum–connectivity co–index  is considered, which generalizes the first zagreb co–index and the general sum–connectivity index of graph theory. we mainly explore the lower and upper bounds in termsof the order and size for this new invariant. additionally, the nordhaus–gaddum–type resultis also represented.

In this paper, we obtain the upper and lower bounds on the eccen- tricity connectivity index of unicyclic graphs with perfect matchings. Also we give some lower bounds on the eccentric connectivity index of unicyclic graphs with given matching numbers.

In this paper we present explicit formulas for the eccentric connectivity index of three classes of chain hexagonal cacti. Further, it is shown that the extremal chain hexagonal cacti with respect to the eccentric connectivity index belong to one of the considered types. Some open problems and possible directions of further research are mentioned in the concluding section.

In theoretical chemistry, -modified Wiener index is a graph invariant topological index to analyze the chemical properties of molecular structure. In this note, we determine the minimum -modified Wiener index of graph with fixed connectivity or edge-connectivity. Our results also present the sufficient and necessary condition for reaching the lower bound.

the stability of a communication network composed of processing nodes and communication links is of prime importance to network designers. as the network begins losing links or nodes, eventually there is a loss in its effectiveness. thus, communication networks must be constructed to be as stable as possible, not only with respect to the initial disruption, but also with respect to the possible...