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The reciprocal degree resistance distance index of a connected graph G is defined as RDRG=∑u,v⊆VGdGu+dGv/rGu,v, where rGu,v is the resistance distance between vertices u and v in G. Let Un denote the set of unicyclic graphs with n vertices. We study the graph with maximum reciprocal degree resistance distance index among all graphs in Un and characterize the corresponding extremal graph.

In this paper, in terms of Wiener index, hyper-Wiener index and Harary index, we first give some sufficient conditions for a nearly balance bipartite graph with given minimum degree to be traceable. Secondly, we establish some conditions for a k-connected graph to be Hamilton-connected and traceable for every vertex, respectively.

Let G be a simple graph and A(G) be the adjacency matrix of G. The eigenvalues of A(G) are referred to as the eigenvalues of G. In this paper, we characterize the graphs with the minimal least eigenvalue among all graphs whose complements are connected and have only two pendent vertices.

In this paper, we establish sufficient conditions for a connected claw-free graph to be traceable in terms of the signless Laplacian spectral radius of the graph or Laplacian eigenvalues of its complements, respectively.

Let S n c be the set of all connected graph each of which is a complement of an n -vertex unicyclic graph. Li and Wang (2012) determined the graphs with the least signless Laplacian eignvalue among all the graphs of the complements of n -vertex trees. In this paper, as a continuance of it, the unique graph among S n c which minimizes the least signless Laplacian eigenvalue is identified.

The hyper-Wiener index WW(G) is defined as with the summation going over all pairs of vertices in G. In this paper, we determine graphs with the minimum hyper-Wiener index among all the unicyclic graphs with n vertices and k pendent vertices.

The Harary index of G is the sum of reciprocals of distance between any two vertices in G. In this paper, we obtain the graphs with the maximum and second-maximum Harary indices among n-vertex unicyclic graphs with diameter d.

In this paper, we study the Hamiltonicity of graphs with large minimum degree. Firstly, we present some conditions for a simple graph to be Hamilton-connected and traceable from every vertex in terms of the spectral radius of the graph or its complement respectively. Secondly, we give the conditions for a nearly balanced bipartite graph to be traceable in terms of spectral radius, signless Laplacian spectral radius of the graph or its quasi-complement respectively., Comment: 21 pages, 2 figures. arXiv admin note: text overlap with arXiv:1602.01033 by other authors