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Edinah K. Gnang

Publications and source records attributed to Edinah K. Gnang.

At least 19 recordsLinked to original sources
arXiv2026

Every tree on $n$ edges decomposes $K_{nx,nx}$ and $K_{2nx+1}$

We prove that every tree on $n$ edges decomposes $K_{nx,nx}$ and $K_{2nx + 1}$ for all positive integers $x$. The said decompositions are obtained by proving that every tree admits a $\vecβ$-labeling (oriented beta-labeling). Our proof employs the polynomial method by identifying trees as functions in the transformation monoid $\mathbb{Z}_n^{\mathbb{Z}_n}$. A proof of the graceful tree conjecture (1967) follows as an immediate consequence of the current result. Finally, we introduce additional algebraic properties derived from the decomposition results.

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arXiv2026

A Proof of the Tree Packing Conjecture

We prove a conjecture of Gyárfás (1976), which asserts that any family of trees $T_1, \dots, T_{n}$ where each $T_k$ has $k$ vertices packs into $K_n$. We do so by translating the decomposition problem into a labeling problem, namely complete labeling. Our proof employs the polynomial method using a functional reformulation of the conjecture.

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arXiv2025

A proof of the Kotzig-Ringel-Rosa Conjecture

In graph theory, a graceful labeling of a graph with m edges is a labeling of its vertices with a subset of the integers ranging from 0 to m inclusive, such that no two vertices share a label, and each edge is uniquely identified by the absolute difference of labels assigned to its endpoints. The Kotzig-Ringel-Rosa conjecture asserts that every tree admits a graceful labeling. We provide a proof of this long standing conjecture via a functional reformulation of the conjecture and a composition lemma.

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arXiv2024

Apportionable matrices and gracefully labelled graphs

To apportion a complex matrix means to apply a similarity so that all entries of the resulting matrix have the same magnitude. We initiate the study of apportionment, both by unitary matrix similarity and general matrix similarity. There are connections between apportionment and classical graph decomposition problems, including graceful labelings of graphs, Hadamard matrices, and equiangluar lines, and potential applications to instantaneous uniform mixing in quantum walks. The connection between apportionment and graceful labelings allows the construction of apportionable matrices from trees. A generalization of the well-known Eigenvalue Interlacing Inequalities using graceful labelings is also presented. It is shown that every rank one matrix can be apportioned by a unitary similarity, but there are $2\x 2$ matrices that cannot be apportioned. A necessary condition for a matrix to be apportioned by unitary matrix is established. This condition is used to construct a set of matrices with nonzero Lebesgue measure that are not apportionable by a unitary matrix.

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arXiv2022

On Partial Differential Encodings of Boolean Functions

We introduce partial differential encodings of Boolean functions as a way of measuring the complexity of Boolean functions. These encodings enable us to derive from group actions non-trivial bounds on the Chow-Rank of polynomials used to specify partial differential encodings of Boolean functions. We also introduce variants of partial differential encodings called partial differential programs. We show that such programs optimally describe important families of polynomials including determinants and permanents. Partial differential programs also enables to quantitively contrast these two families of polynomials. Finally we derive from polynomial constructions inspired by partial differential programs which exhibit an unconditional exponential separation between high order hypergraph isomorhism instances and their sub-isomorphism counterparts.

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arXiv2020

On graceful labelings of trees

We prove via a composition lemma, the Kotzig-Ringel-Rosa conjecture, better known as the Graceful Labeling Conjecture. We also prove via a stronger version of the composition lemma a stronger form of the Graceful Labeling Conjecture.

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arXiv2020

Growing Graceful and Harmonious Trees

We describe symbolic constructions for listing and enumerating graphs having the same induced edge label sequence. We settle in the affirmative R. Whitty's [W08] conjectured existence of determinantal constructions for listing and enumerating gracefully labeled trees. We conclude the paper with a description of a new graceful labeling algorithm.

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arXiv2019

On the Bhattacharya-Mesner rank of third order hypermatrices

We introduce the Bhattacharya-Mesner rank of third order hypermatrices as a relaxation to the tensor rank and devise from it some bounds for the tensor rank. We use the Bhattacharya-Mesner rank to extend to third order hypermatrices the connection relating the rank to a notion of linear dependence. We also derive explicit necessary and sufficient conditions for the existence of third order hypermatrix inverse pair. Finally we use inverse pair to extend to third order hypermatrices the formulation and proof of the matrix rank-nullity theorem.

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arXiv2019

Spectral Analysis for Non-Hermitian Matrices and Directed Graphs

We generalize classical results in spectral graph theory and linear algebra more broadly, from the case where the underlying matrix is Hermitian to the case where it is non-Hermitian. New admissibility conditions are introduced to replace the Hermiticity condition. We prove new variational estimates of the Rayleigh quotient for non-Hermitian matrices. As an application, a new Delsarte-Hoffman-type bound on the size of the largest independent set in a directed graph is developed. Our techniques consist in quantifying the impact of breaking the Hermitian symmetry of a matrix and are broadly applicable.

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arXiv2015

Approximating the spectrum of matrices and hypermatrices

We describe a general approach for computing generators for elimination ideals associated with matrix and hypermatrix spectral decomposition constraints. We derive from these generators iterative procedures for approximating the spectral decomposition of matrices and hypermatrices.

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arXiv2015

A combinatorial approach to the algebra of hypermatrices

We present two hypermatrix formulations of the Cayley Hamilton theorem. One of the proposed formulation naturally extends to hypermatrices the combinatorial interpretations of the classical Cayley Hamilton theorem. We conclude by discussing an application of the theorem to computing graph invariants which distinguish some non-isomorphic graphs with isospectral adjacency matrices.

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arXiv2014

Counting arithmetic formulas

An arithmetic formula is an expression involving only the constant $1$, and the binary operations of addition and multiplication, with multiplication by $1$ not allowed. We obtain an asymptotic formula for the number of arithmetic formulas evaluating to $n$ as $n$ goes to infinity, solving a conjecture of E. K. Gnang and D. Zeilberger. We give also an asymptotic formula for the number of arithmetic formulas evaluating to $n$ and using exactly $k$ multiplications. Finally we analyze three specific encodings for producing arithmetic formulas. For almost all integers $n$, we compare the lengths of the arithmetic formulas for $n$ that each encoding produces with the length of the shortest formula for $n$ (which we estimate from below). We briefly discuss the time-space tradeoff offered by each.

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