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Let G be a connected graph with minimum degree δ(G) and vertexconnectivity κ(G). The graph G is k-connected if κ(G) ≥ k, maximally connected if κ(G) = δ(G), and super-connected (or super-κ) if every minimum vertex-cut isolates a vertex of minimum degree. In this paper, we show that a connected graph or a connected triangle-free graph is k-connected, maximally connected or superconnected if the number of edges or the spectral radius is large enough.

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A pebbling move on a graph G consists of the removal of two pebbles from one vertex and the placement of one pebble on an adjacent vertex. Rubbling is a version of pebbling where an additional move is allowed, which is also called the strict rubbling move. In this new move, one pebble each is removed from u and v adjacent to a vertex w , and one pebble is added on w . The rubbling number of a graph G is the smallest number m , such that one pebble can be moved to each vertex from every distribution with m pebbles. The optimal rubbling number of a graph G is the smallest number m , such that one pebble can be moved to each vertex from some distribution with m pebbles. In this paper, we give short proofs to determine the rubbling number of cycles and the optimal rubbling number of paths, cycles, and the grid P 2 × P n ; moreover, we give an upper bound of the optimal rubbling number of P m × P n .

The definition of a Detour–Harary index is ω H ( G ) = 1 2 ∑ u , v ∈ V ( G ) 1 l ( u , v | G ) , where G is a simple and connected graph, and l ( u , v | G ) is equal to the length of the longest path between vertices u and v. In this paper, we obtained the maximum Detour–Harary index about unicyclic graphs, bicyclic graphs, and cacti, respectively.