The forgotten topological index of a (molecule) graph is the sum of cubes of all its vertex degrees, which plays a significant role in measuring the branching of the carbon atom skeleton. It is meaningful and difficult to explore sufficient conditions for a given graph keeping certain properties in graph theory. In this paper, we mainly explore sufficient conditions in terms of the forgotten topological index for a graph to be ℓ-connected, ℓ-deficient, ℓ-Hamiltonian and ℓ⁻-independent, respectively. The conditions cannot be dropped.

Graph invariants, based on the distances between the vertices of a graph, are widely used in theoretical chemistry. Recently, Gutman, Feng and Yu (Transactions on Combinatorics, 01 (2012) 27- 40) introduced the degree resistance distance of a graph G, which is defined as DR(G) = Σ{u,v}V(G)[dG(u)+dG(v)]RG(u,v), where dG(u) is the degree of vertex u of the graph G, and RG(u, v) denotes the resistance distance between the vertices u and v of the graph G. Further, they characterized n-vertex unicyclic graphs having minimum and second minimum degree resistance distance. In this paper, we characterize n-vertex bicyclic graphs having maximum degree resistance distance.

Each traceable graph must be a block-chain; however, a block-chain is not necessarily traceable in general. Whether a given graph is a block-chain or not can be easily verified by a polynomial algorithm. It occurs to us that forbidden subgraph conditions for a block-chain are traceable. In this article, we characterize all pairs of disconnected forbidden subgraphs for the traceability of block-chains, so as to completely solve pairs of forbidden subgraphs for the traceability of block-chains (including disconnected and connected).