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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.

A graph is called unicyclic if the graph contains exactly one cycle. Unicyclic graphs with the fourth extremal Wiener indices are characterized. It is shown that, among all unicyclic graphs with n≥8 vertices, C5Sn−4 and C2u1,u2S3,Sn−4 attain the fourth minimum Wiener index, whereas C3u1,u2P3,Pn−4 attains the fourth maximum Wiener index.

The Merrifield–Simmons index i G of a graph G is defined as the number of subsets of the vertex set, in which any two vertices are nonadjacent, i.e., the number of independent vertex sets of G . In this paper, we determine the minimum Merrifield–Simmons index of unicyclic graphs with n vertices and diameter at most four.

In this work, by using the properties of the variable sum exdeg indices and analyzing the structure of the quasi-tree graphs and unicyclic graphs, the minimum and maximum variable sum exdeg indices of quasi-tree graphs and quasi-tree graphs with perfect matchings were presented; the minimum and maximum variable sum exdeg indices of unicyclic graphs with given pendant vertices and cycle length were determined.

The variable connectivity index, introduced by the chemist Milan Randić in the first quarter of 1990s, for a graph G is defined as ∑vw∈EGdv+γdw+γ−1/2, where γ is a non-negative real number and dw is the degree of a vertex w in G. We call this index as the variable Randić index and denote it by Rvγ. In this paper, we show that the graph created from the star graph of order n by adding an edge has the minimum Rvγ value among all unicyclic graphs of a fixed order n, for every n≥4 and γ≥0.

In this paper, we study the properties and structure of the maximal-adjacency-spectrum unicyclic graphs with given maximum degree. We obtain some necessary conditions on the maximal-adjacency-spectrum unicyclic graphs in the set of unicyclic graphs with n vertices and maximum degree Δ and describe the structure of the maximal-adjacency-spectrum unicyclic graphs in the set. Besides, we also give a new upper bound on the adjacency spectral radius of unicyclic graphs, and this new upper bound is the best upper bound expressed by vertices n and maximum degree Δ from now on.

Graphs are used to model chemical compounds and drugs. In the graphs, each vertex represents an atom of molecule and edges between the corresponding vertices are used to represent covalent bounds between atoms. TheWiener polarity indexWp(G)of a graphGis the number of unordered pairs of verticesu,vofGsuch that the distance betweenuandvis equal to 3. The trees and unicyclic graphs with perfect matching, of which all vertices have degrees not greater than three, are referred to as theHückel treesandunicyclic Hückel graphs, respectively. In this paper, we first consider the smallest and the largest Wiener polarity index among all Hückel trees on2nvertices and characterize the corresponding extremal graphs. Then we obtain an upper and lower bound for the Wiener polarity index of unicyclic Hückel graphs on2nvertices.

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.

The aim of this report to solve the open problem suggested by Chen et al. We study the graph entropy with ABC edge weights and present bounds of it for connected graphs, regular graphs, complete bipartite graphs, chemical graphs, tree, unicyclic graphs, and star graphs. Moreover, we compute the graph entropy for some families of dendrimers.

If G is a simple connected graph with vertex V(G), then the eccentric distance sum of G, denoted by ξd(G), is defined as ∑v∈V(G)ecG(v)DG(v), where ecG(v) is the eccentricity of the vertex v and DG(v) is the sum of all distances from the vertex v. Let n≥8. We determine the n-vertex trees with, respectively, the maximum, second-maximum, third-maximum, and fourth-maximum eccentric distance sums. We also characterize the extremal unicyclic graphs on n vertices with respectively, the maximal, second maximal, and third maximal eccentric distance sums.

The absolute sum of chromatic polynomial coefficient of forest,q-tree, unicyclic graphs, and quasiwheel graphs, are determined in this paper.

The Wiener polarity index of a connected graphGis defined as the number of its pairs of vertices that are at distance three. By introducing some graph transformations, in different way with that of Huang et al., 2013, we determine the minimum Wiener polarity index of unicyclic graphs with any given maximum degree and girth, and characterize extremal graphs. These observations lead to the determination of the minimum Wiener polarity index of unicyclic graphs and the characterization of the extremal graphs.

LetW(n;q,m1,m2)be the unicyclic graph withnvertices obtained by attaching two paths of lengthsm1andm2at two adjacent vertices of cycleCq. LetU(n;q,m1,m2,…,ms)be the unicyclic graph withnvertices obtained by attachingspaths of lengthsm1,m2,…,msat the same vertex of cycleCq. In this paper, we prove thatW(n;q,m1,m2)andU(n;q,m1,m2,…,ms)are determined by their Laplacian spectra whenqis even.