Decomposition theorem on matchable distributive lattices

A distributive lattice structure ${\mathbf M}(G)$ has been established on the set of perfect matchings of a plane bipartite graph $G$. We call a lattice {\em matchable distributive lattice} (simply MDL) if it is isomorphic to such a distributive lattice. It is natural to ask which lattices are MDLs. We show that if a plane bipartite graph $G$ is elementary, then ${\mathbf M}(G)$ is irreducible. Based on this result, a decomposition theorem on MDLs is obtained: a finite distributive lattice $\mathbf{L}$ is an MDL if and only if each factor in any cartesian product decomposition of $\mathbf{L}$ is an MDL. Two types of MDLs are presented: $J(\mathbf{m}\times \mathbf{n})$ and $J(\mathbf{T})$, where $\mathbf{m}\times \mathbf{n}$ denotes the cartesian product between $m$-element chain and $n$-element chain, and $\mathbf{T}$ is a poset implied by any orientation of a tree.Comment: 19 pages, 7 figure

Maximum cardinality resonant sets and maximal alternating sets of hexagonal systems

AbstractIt is shown that the Clar number can be arbitrarily larger than the cardinality of a maximal alternating set. In particular, a maximal alternating set of a hexagonal system need not contain a maximum cardinality resonant set, thus disproving a previously stated conjecture. It is known that maximum cardinality resonant sets and maximal alternating sets are canonical, but the proofs of these two theorems are analogous and lengthy. A new conjecture is proposed and it is shown that the validity of the conjecture allows short proofs of the aforementioned two results. The conjecture holds for catacondensed hexagonal systems and for all normal hexagonal systems up to ten hexagons. Also, it is shown that the Fries number can be arbitrarily larger than the Clar number

Forcing, Freedom, & Uniqueness in Graph Theory & Chemistry

Harary’s & Randić’s ideas of “forcing” & “freedom” involve subsets of double bonds of Kekule structure such as to be unique to that Kekule structure. Such forcing sets are argued to be greatly generalizable to deal with various other coverings, and thence forcing seems to be fundamental, and of notable potential utility. Various forcing invariants associated to (molecular) graphs ensue, with illustrative (chemical) ex-amples and some mathematical consequences being provided. A complementary “uniqueness” idea is not-ed, and the general characteristic of “derivativity” of “forcing” is established (as is relevant for QSPR fit-tings). Different ways in which different sorts of forcings arise in chemistry are briefly indicated.(doi: 10.5562/cca2000

Ordering of minimal energies in unicyclic signed graphs

Let S = (G, σ) be a signed graph of order n and size m and let t1, t2, . . . , tn be the eigenvalues of S. The energy of S is defined as E(S) = Pnj=1|tj|. A connected signed graph is said to be unicyclic if its order and size are same. In this paper, we characterize, up to switching, theunicyclic signed graphs with first 11 minimal energies for all n ≥ 12. For 3 ≤ n ≤ 7, we provide complete ordering of unicyclic signed graphs with respect to energy. For n = 8, 9, 10 and 11, we determine unicyclic signed graphs with first 13 minimal energies respectively