The Hosoya index and the Merrifield–Simmons index of a graph are defined as the total number of the matchings (including the empty edge set) and the total number of the independent vertex sets ...(including the empty vertex set) of the graph, respectively. Let
V
n
,
k
be the set of connected
n
-vertex graphs with connectivity at most
k
. In this note, we characterize the extremal (maximal and minimal) graphs from
V
n
,
k
with respect to the Hosoya index and the Merrifield–Simmons index, respectively.
By using the Z-index a useful algorithm for obtaining such "dormant graphs" was discovered, which systematically generates pairs of isospectral tree graphs, although mathematical rigorousness is not ...yet completely attained.
Given a graph G=(V,E), let the triangulationG△=(V△,E△)ofG be the graph obtained from G by supplementing each uv∈E with a new vertex w along with new edges uw and wv (while retaining uv). Let dv be ...the degree of a vertex v∈V and let G be a tree T. Then it is proved that the count of perfect matchings of the Cartesian product of T△ with K2 is given as the product of factors dv+1 over all v∈V. Also under favorable conditions, the degree sequence of T△×K2 is reconstructed via factorization of the number of its perfect matchings. Previously introduced degree product polynomials play a helpful role.
The personal history of the present author's research career of almost forty-five years in the application of graph theory to chemistry is briefly described in an essay with particular reference to ...the author's "bright side of mathematical chemistry." The chosen topics are as follows: two seminal papers by the present author, more than 1500 citations of his topological index, Z, Hosoya polynomial, Hosoya items, more than 200 papers carrying "Hosoya" in the title, Erdös number, etc.
By using the Z-index a useful algorithm for obtaining such "dormant graphs" was discovered, which systematically generates pairs of isospectral tree graphs, although mathematical rigorousness is not ...yet completely attained.
The Hosoya index and the Merrifield–Simmons index of a graph are defined as the total number of the matchings (including the empty edge set) and the total number of the independent vertex sets ...(including the empty vertex set) of the graph, respectively. Let
W
n
,
k
be the set of connected graphs with
n
vertices and clique number
k
. In this note we characterize the graphs from
W
n
,
k
with extremal (maximal and minimal) Hosoya indices and the ones with extremal (maximal and minimal) Merrifield–Simmons indices, respectively.
The unicyclic graphs with perfect matchings having degrees not greater than three are referred to as the unicyclic Hückel graphs. The set of these graphs with
2
n
vertices is denoted by
H
2
n
. By a ...new method proposed here, we obtain the first 7 graphs in the increasing order of their Hosoya indices within
H
2
n
for
n
≥
8
. In addition, complete increasing orders of graphs in terms of their Hosoya indices are also derived for four subsets in
H
2
n
.
The Merrifield-Simmons index of a graph is defined as the total number of the independent sets of the graph and the Ho- soya index of a graph is defined as the total number of the match- ings of the ...graph. In this paper, the definition of a class of po- lygonal chains is given, ordering of the polygonal chains with respect to Merrifield-Simmons index and Hosoya index are ob- tained, and their extremal graphs with respect to these two topo- logical indices are determined.
It is well known that the two graph invariants, “the Merrifield–Simmons index” and “the Hosoya index” are important in structural chemistry. A graph
G
is called a quasi-tree graph, if there exists
u
...0
in
V
(
G
)
such that
G
−
u
0
is a tree. In this paper, at first we characterize the
n
-vertex quasi-tree graphs with the largest, the second-largest, the smallest and the second-smallest Merrifield–Simmons indices. Then we characterize the
n
-vertex quasi-tree graphs with the largest, the second-largest, the smallest and the second-smallest Hosoya indices, as well as those
n
-vertex quasi-tree graphs with
k
pendent vertices having the smallest Hosoya index.
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