7 research outputs found
Mathematical aspects of Wiener index
The Wiener index (i.e., the total distance or the transmission number),
defined as the sum of distances between all unordered pairs of vertices in a
graph, is one of the most popular molecular descriptors. In this article we
summarize some results, conjectures and problems on this molecular descriptor,
with emphasis on works we were involved in.Comment: 28 pages, 4 figures, survey pape
A survey of recent results in (generalized) graph entropies
The entropy of a graph was first introduced by Rashevsky \cite{Rashevsky} and
Trucco \cite{Trucco} to interpret as the structural information content of the
graph and serve as a complexity measure. In this paper, we first state a number
of definitions of graph entropy measures and generalized graph entropies. Then
we survey the known results about them from the following three respects:
inequalities and extremal properties on graph entropies, relationships between
graph structures, graph energies, topological indices and generalized graph
entropies, complexity for calculation of graph entropies. Various applications
of graph entropies together with some open problems and conjectures are also
presented for further research.Comment: This will appear as a chapter "Graph Entropy: Recent Results and
Perspectives" in a book: Mathematical Foundations and Applications of Graph
Entrop
Extremal graphs for vertex-degree-based invariants with given degree sequences
For a symmetric bivariable function , let the {\it connectivity
function} of a connected graph be ,
where is the degree of vertex . In this paper, we prove that for an
escalating (de-escalating) function , there exists a BFS-graph with the
maximum (minimum) connectivity function among all graphs with a
cyclic degree sequence and , and obtain
the majorization theorem for connectivity function for unicyclic and bicyclic
degree sequences. Moreover, some applications of graph invariants based on
degree are included.Comment: 23 page
On Minimum Wiener Polarity Index of Unicyclic Graphs with Prescribed Maximum Degree
The Wiener polarity index of a connected graph G is 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