Related papers: Equivalence of the Generalized Zhang-Zhang Polynom…
The Clar covering polynomial (also called Zhang-Zhang polynomial in some chemical literature) of a hexagonal system is a counting polynomial for some types of resonant structures called Clar covers, which can be used to determine Kekul\'e…
We report a closed-form formula for the Zhang-Zhang polynomial (aka ZZ polynomial or Clar covering polynomial) of an important class of elementary pericondensed benzenoids $Rb\left(n_{1},n_{2},m_{1},m_{2}\right)$ usually referred to as…
Let $G$ be a graph embedded in a surface and let $\mathcal F$ be a set of even faces of $G$ (faces bounded by a cycle of even length). The resonance graph of $G$ with respect to $\mathcal F$, denoted by $R(G;\mathcal F)$, is a graph such…
Klav\v{z}ar and Mollard introduced daisy cubes which are interesting isometric subgraphs of $ n$-cubes $Q_n$, induced with intervals between the maximal elements of a poset $ (V (Q_n),\leq)$ and the vertex $ 0^n \in V (Q_n)$. In this paper…
In this paper we generalize the binary coding procedure of perfect matchings from catacondensed benzenoid graphs to catacondensed even ring systems (also called cers). Next, we study cers with isomorphic resonance graphs. For this purpose,…
Partial cubes are the graphs which can be embedded into hypercubes. The {\em cube polynomial} of a graph $G$ is a counting polynomial of induced hypercubes of $G$, which is defined as $C(G,x):=\sum_{i\geqslant 0}\alpha_i(G)x^i$, where…
Benzene is a hexagonal molecule of six carbon atoms, each of which is bound to six hydrogen atoms. The equality of all six CC bond lengths, despite the alternating double and single bonds, and the surplus (resonance) energy, led to the…
A catacondensed even ring system (shortly CERS) is a simple bipartite 2-connected outerplanar graph with all vertices of degree 2 or 3. In this paper, we investigate the resonance graphs (also called $Z$-transformation graphs) of CERS and…
The type-PQ adjacency polytope associated to a simple graph is a $0/1$-polytope containing valuable information about an underlying power network. Chen and the first author have recently demonstrated that, when the underlying graph $G$ is…
Using the orthonormality of the 2D-Zernike polynomials, reproducing kernels, reproducing kernel Hilbert spaces, and ensuring coherent states attained. With the aid of the so-obtained coherent states, the complex unit disc is quantized.…
While motivated by structural problems in mathematical music theory, this article introduces a novel combinatorial framework that advances the classification of cyclic cubic bipartite graphs. We extend the classical study of Levi graphs by…
In this paper, we consider the plasmon resonance in multi-layer structures. The conductivity problem associated with uniformly distributed background field is considered. We show that the plasmon mode is equivalent to the eigenvalue problem…
The article studies power complexes and generalized power complexes, and investigates the algebraic structure of their automorphism groups. The combinatorial incidence structures involved are cube-like, in the sense that they have many…
The cyclic sieving phenomenon provides a link between a polynomial analogue of Gauss congruence known as $q$-Gauss congruence, and a combinatorial analogue of Gauss congruence based on sequences of cyclic group actions. We strengthen this…
Matrix congruence can be used to mimic linear maps between homogeneous quadratic polynomials in $n$ variables. We introduce a generalization, called standard-form congruence, which mimics affine maps between non-homogeneous quadratic…
In this thesis we studied the structure coefficients and especially their dependence on $n$ in the case of a sequence of double-class algebras. The first chapter is dedicated to the study of the structure coefficients in the general cases…
We establish $q$-analogs for four congruences involving central binomial coefficients. The $q$-identities necessary for this purpose are shown via the $q$-WZ method.
Determination of topological invariants of graphene flakes, nanotubes, and fullerenes constitutes a challenging task due to its time-intensive nature and exponential scaling. The invariants can be organized in a form of a combinatorial…
This paper investigates the interplay between local and global equivalences on noncommutative polynomials, the elements of the free algebra. When the latter are viewed as functions in several matrix variables, a local equivalence of…
Individual Kekule valence structures of biphenylene and related hydrocarbons are treated perturbatively by modelling them as sets of weakly-interacting uniform double bonds. Total pi-electron energies of these structures are then expressed…