21 research outputs found
Two Classes of Topological Indices of Phenylene Molecule Graphs
A phenylene is a conjugated hydrocarbons molecule composed of six- and four-membered rings. The matching energy of a graph G is equal to the sum of the absolute values of the zeros of the matching polynomial of G, while the Hosoya index is defined as the total number of the independent edge sets of G. In this paper, we determine the extremal graph with respect to the matching energy and Hosoya index for all phenylene chains
Lower matching conjecture, and a new proof of Schrijver's and Gurvits's theorems
Friedland's Lower Matching Conjecture asserts that if is a --regular
bipartite graph on vertices, and denotes the number of
matchings of size , then where . When
, this conjecture reduces to a theorem of Schrijver which says that a
--regular bipartite graph on vertices has at least
perfect matchings. L. Gurvits
proved an asymptotic version of the Lower Matching Conjecture, namely he proved
that
In this paper, we prove the Lower Matching Conjecture. In fact, we will prove
a slightly stronger statement which gives an extra factor
compared to the conjecture if is separated away from and , and is
tight up to a constant factor if is separated away from . We will also
give a new proof of Gurvits's and Schrijver's theorems, and we extend these
theorems to --biregular bipartite graphs