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An even factor of a graph G is a spanning subgraph in G such that the degree of each vertex is a positive even integer. In this paper, we show that for any induced claw of simple graph G of order at least 10, if there exists at least a pair of vertices out of the claw such that they are the common neighbors of nonadjacent vertices of the claw, then G has an even factor if and only if $$\delta (G)\ge 2$$ and every odd branch-bond of G contains a branch of length 1. The even factor of claw-heavy graphs was also characterized.

Let $$G$$G be a connected graph with order $$n,$$n, minimum degree $$\delta =\delta (G)$$?=?(G) and edge-connectivity $$\lambda =\lambda (G).$$?=?(G). A graph $$G$$G is maximally edge-connected if $$\lambda =\delta .$$?=?. In this paper, we present two sufficient conditions for graphs to be maximally edge-connected, which generalize two results recently proved by P. Dankelmann, A. Hellwig and L. Volkmann. The extremal graphs are also characterized.

In this contribution, we first investigate sharp bounds for the reciprocal sum-degree distance of graphs with a given matching number. The corresponding extremal graphs are characterized completely. Then we explore the $$k$$k-decomposition for the reciprocal sum-degree distance. Finally, we establish formulas for the reciprocal sum-degree distance of join and the Cartesian product of graphs.