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.
LetGbe an undirected simple graph of ordern. LetA(G)be the adjacency matrix ofG, and letμ1(G)≤μ2(G)≤⋯≤μn(G)be its eigenvalues. The energy ofGis defined asℰ(G)=∑i=1n|μi(G)|. Denote byGBPTa bipartite graph. In this paper, we establish the sufficient conditions forGhaving a Hamiltonian path or cycle or to be Hamilton-connected in terms of the energy of the complement ofG, and give the sufficient condition forGBPThaving a Hamiltonian cycle in terms of the energy of the quasi-complement ofGBPT.
The hyper-Wiener index WW(G) is defined asthe 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 hyper-Wiener index is a kind of extension of the Wiener index, used for predicting physicochemical properties of organic compounds. The hyper-Wiener indexWW(G)is defined asWW(G)=1/2∑u,v∈VGdGu,v+dG2u,vwith the summation going over all pairs of vertices inG, anddGu,vdenotes the distance of the two verticesuandvin the graphG. In this paper, we obtain the second-minimum hyper-Wiener indices among all the trees withnvertices and diameterdand characterize the corresponding extremal graphs.
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