The Hosoya index of a graph is defined by the total number of the matchings of the graph. In this paper, we determine the maximum Hosoya index of unicyclic graphs with n vertices and diameter 3 or 4. ...Our results somewhat answer a question proposed by Wagner and Gutman in 2010 for unicyclic graphs with small diameter.
A vast amount of information about distance based graph invariants is contained in the Hosoya polynomial. Such an information is helpful to determine well-known distance based molecular descriptors. ...The Hosoya index or
-index of a graph
is the total number of its matching. The Hosoya index is a prominent example of topological indices, which are of great interest in combinatorial chemistry, and later on it applies to address several chemical properties in molecular structures. In this article, we investigate Hosoya properties (Hosoya polynomial, reciprocal Hosoya polynomial and Hosoya index) of the commuting graph associated with an algebraic structure developed by the symmetries of regular molecular gones (constructed by atoms with regular atomic-bonding).
The Hosoya index and the Merrifield-Simmons index are typical examples of graph invariants used in mathematical chemistry for quantifying relevant details of molecular structure. In recent years, ...quite a lot of work has been done on the extremal problem for these two indices, i.e., the problem of determining the graphs within certain prescribed classes that maximize or minimize the index value. This survey collects and classifies these results, and also provides some useful auxiliary results, tools and techniques that are frequently used in the study of this type of problem.
The Hosoya index m(G) and the Merrifield–Simmons index i(G) of a graph G are the number of matchings and the number of independent sets in G. In this paper, we establish exact formulas for the ...expected value of the Hosoya index and Merrifield–Simmons index of the random cyclooctylene chains, which are graphs of a chemical chain consisting of n octagons, each of which is connected to the end of the previous octagon by an edge. In addition, we obtain the expected values and the average values of the two indexes through the relevant chemical diagrams and a series of accurate formulas with respect to the set of all cyclooctylene chains with n octagons.
The Hosoya index of a graph is defined as the total number of its independent edge sets. This index is an important example of topological indices, a molecular-graph based structure descriptor that ...is of significant interest in combinatorial chemistry. The Hosoya index inspires the introduction of a matrix associated with a molecular acyclic graph called the Hosoya matrix. We propose a simple linear-time algorithm, which does not require pre-processing, to compute the Hosoya index of an arbitrary tree. A similar approach allows us to show that the Hosoya matrix can be computed in constant time per entry of the matrix.
In this article, we give sharp bounds on the Hosoya index and the Merrifield–Simmons index for connected graphs of fixed size. As a consequence, we determine all connected graphs of any fixed order ...and size which maximize the Merrifield–Simmons index. Sharp lower bounds on the Hosoya index are known for graphs of order
n
and size
m
∈
n
-
1
,
2
n
-
3
∪
n
-
1
2
,
n
2
; while sharp upper bounds were only known for graphs of order
n
and size
m
≤
n
+
2
. We give sharp upper bounds on the Hosoya index for dense graphs with
m
≥
n
2
-
2
n
/
3
. Moreover, all extreme graphs are also determined.
We consider an earlier much studied quasi-order, defined in terms of matching numbers of graphs, and apply it to graph complements. We establish four transformations on the complements of graphs that ...increase or decrease the matching numbers accordingly. Several applications of these results are put forward.
The study of the invariance of the skew spectrum of an oriented graph under different orientations leads to the special family G of graphs without even cycles. Basic graphical and spectral properties ...of G are observed. As a result, the Hosoya index of a graph in G can be computed via its skew spectrum. We also focus on friendship graphs, a subfamily of G. In particular, the condition under which the skew spectrum of a friendship graph is actually i times the spectrum of some graph is determined, where i=−1.
Let G be a graph and Z(G) be its Hosoya index. We show how the Hosoya index can be a good tool to establish some new identities involving Fibonacci numbers. This permits to extend Hillard and ...Windfeldt work.
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