The Wiener index of a graph is the sum of the distances between all pairs of vertices. It has been one of main descriptors that correlate a chemical compound's molecular graph with experimentally gathered data regarding the compound's characteristics. We characterize graphs with the maximum Wiener index among all graphs of order . with radius two. In addition, we pose a conjecture concerning the minimum Wiener index of graphs with given radius. If this conjecture is true, it will be able to answer an open question by You and Liu (2011).

The linear arboricity la(G) of a graph G is the minimum number of linear forests which partition the edges of G. An and Wu introduce the notion of list linear arboricity lla(G) of a graph G and conjecture that lla(G) = la(G) for any graph G. We confirm that this conjecture is true for any planar graph having ∆ 13, or for any planar graph with ∆ 7 and without i-cycles for some i ∈ {3, 4, 5}. We also prove that ∆(G) 2 lla(G) ∆(G)+1 2 for any planar graph having ∆ 9.

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The group coloring of graphs is a new kind of graph coloring, introduced by Jaeger et al. in 1992, and the group chromatic number of a graph G is denoted by χg(G). In this note, we prove that for a positive integer k, a graph G with χg(G) > k can be obtained from any complete bipartite graph G 0 with χg(G 0 ) > k by certain types of graph operations.