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Jephian C. -H. Lin

Publications and source records attributed to Jephian C. -H. Lin.

34 records · Page 2Linked to original sources
arXiv2019

Comparability and Cocomparability Bigraphs

We propose bipartite analogues of comparability and cocomparability graphs. Surprizingly, the two classes coincide. We call these bipartite graphs cocomparability bigraphs. We characterize cocomparability bigraphs in terms of vertex orderings, forbidden substructures, and orientations of their complements. In particular, we prove that cocomparability bigraphs are precisely those bipartite graphs that do not have edge-asteroids; this is analogous to Gallai's structural characterization of cocomparability graphs by the absence of (vertex-) asteroids. Our characterizations imply a robust polynomial-time recognition algorithm for the class of cocomparability bigraphs. Finally, we also discuss a natural relation of cocomparability bigraphs to interval containment bigraphs, resembling a well-known relation of cocomparability graphs to interval graphs.

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arXiv2018

Properties of a $q$-analogue of zero forcing

Zero forcing is a combinatorial game played on a graph where the goal is to start with all vertices unfilled and to change them to filled at minimal cost. In the original variation of the game there were two options. Namely, to fill any one single vertex at the cost of a single token; or if any currently filled vertex has a unique non-filled neighbor, then the neighbor is filled for free. This paper investigates a $q$-analogue of zero forcing which introduces a third option involving an oracle. Basic properties of this game are established including determining all graphs which have minimal cost $1$ or $2$ for all possible $q$, and finding the zero forcing number for all trees when $q=1$.

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arXiv2018

Rigid linkages and partial zero forcing

Connections between vital linkages and zero forcing are established. Specifically, the notion of a rigid linkage is introduced as a special kind of unique linkage and it is shown that spanning forcing paths of a zero forcing process form a spanning rigid linkage and thus a vital linkage. A related generalization of zero forcing that produces a rigid linkage via a coloring process is developed. One of the motivations for introducing zero forcing is to provide an upper bound on the maximum multiplicity of an eigenvalue among the real symmetric matrices described by a graph. Rigid linkages and a related notion of rigid shortest linkages are utilized to obtain bounds on the multiplicities of eigenvalues of this family of matrices.

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arXiv2018

The sepr-sets of sign patterns

Given a real symmetric $n\times n$ matrix, the sepr-sequence $t_1\cdots t_n$ records information about the existence of principal minors of each order that are positive, negative, or zero. This paper extends the notion of the sepr-sequence to matrices whose entries are of prescribed signs, that is, to sign patterns. A sufficient condition is given for a sign pattern to have a unique sepr-sequence, and it is conjectured to be necessary. The sepr-sequences of sign semi-stable patterns are shown to be well-structured; in some special circumstances, the sepr-sequence is enough to guarantee the sign pattern being sign semi-stable. In alignment with previous work on symmetric matrices, the sepr-sequences for sign patterns realized by symmetric nonnegative matrices of orders two and three are characterized.

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arXiv2018

On the distance matrices of the CP graphs

This paper introduces a new class of graphs, the CP graphs, and shows that their distance determinant and distance inertia are independent of their structures. The CP graphs include the family of linear $2$-trees. When a graph is attached with a CP graph, it is shown that the distance determinant and the distance inertia are also independent of the structure of the CP graph. Applications to the addressing problem proposed by Graham and Pollak in 1971 are given.

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arXiv2017

Critical ideals, minimum rank and zero forcing number

There are profound relations between the zero forcing number and minimum rank of a graph. We study the relation of both parameters with a third one, the algebraic co-rank; that is defined as the largest $i$ such that the $i$-th critical ideal is trivial. This gives a new perspective for bounding and computing these three graph parameters.

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arXiv2017

On the error of a priori sampling: zero forcing sets and propagation time

Zero forcing is an iterative process on a graph used to bound the maximum nullity. The process begins with select vertices as colored, and the remaining vertices can become colored under a specific color change rule. The goal is to find a minimum set of vertices such that after iteratively applying the rule, all of the vertices become colored (i.e., a minimum zero forcing set). Of particular interest is the propagation time of a chosen set which is the number of steps the rule must be applied in order to color all the vertices of a graph. We give a purely linear algebraic interpretation of zero forcing: Find a set of vertices $S$ such that for any weighted adjacency matrix $\mathbf{A}$, whenever $\mathbf{Ax} = \mathbf{0}$, the entirety of of $\mathbf{x}$ can be recovered using only $\mathbf{x}_S$, the entries corresponding to $S$. The key here is that $S$ must be chosen before $\mathbf{A}$. In this light, we are able to give a linear algebraic interpretation of the propagation time: Any error in $\mathbf{x}_S$ effects the error of $\mathbf{x}$ exponentially in the propagation time. This error can be quantitatively measured using newly defined zero forcing-related parameters, the error polynomial vector and the variance polynomial vector. In this sense, the quality of two zero forcing sets can objectively be compared even if the sets are the same size and their propagation time is the same. Examples and constructions are given.

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arXiv2017

Analogies between the crossing number and the tangle crossing number

Tanglegrams are special graphs that consist of a pair of rooted binary trees with the same number of leaves, and a perfect matching between the two leaf-sets. These objects are of use in phylogenetics and are represented with straightline drawings where the leaves of the two plane binary trees are on two parallel lines and only the matching edges can cross. The tangle crossing number of a tanglegram is the minimum crossing number over all such drawings and is related to biologically relevant quantities, such as the number of times a parasite switched hosts. Our main results for tanglegrams which parallel known theorems for crossing numbers are as follows. The removal of a single matching edge in a tanglegram with $n$ leaves decreases the tangle crossing number by at most $n-3$, and this is sharp. Additionally, if $γ(n)$ is the maximum tangle crossing number of a tanglegram with $n$ leaves, we prove $\frac{1}{2}\binom{n}{2}(1-o(1))\leγ(n)<\frac{1}{2}\binom{n}{2}$. Further, we provide an algorithm for computing non-trivial lower bounds on the tangle crossing number in $O(n^4)$ time. This lower bound may be tight, even for tanglegrams with tangle crossing number $Θ(n^2)$.

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arXiv2017

The inverse eigenvalue problem of a graph: Multiplicities and minors

The inverse eigenvalue problem of a given graph $G$ is to determine all possible spectra of real symmetric matrices whose off-diagonal entries are governed by the adjacencies in $G$. Barrett et al. introduced the Strong Spectral Property (SSP) and the Strong Multiplicity Property (SMP) in [8]. In that paper it was shown that if a graph has a matrix with the SSP (or the SMP) then a supergraph has a matrix with the same spectrum (or ordered multiplicity list) augmented with simple eigenvalues if necessary, that is, subgraph monotonicity. In this paper we extend this to a form of minor monotonicity, with restrictions on where the new eigenvalues appear. These ideas are applied to solve the inverse eigenvalue problem for all graphs of order five, and to characterize forbidden minors of graphs having at most one multiple eigenvalue.

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arXiv2017

Zero forcing number, Grundy domination number, and their variants

This paper presents strong connections between four variants of the zero forcing number and four variants of the Grundy domination number. These connections bridge the domination problem and the minimum rank problem. We show that the Grundy domination type parameters are bounded above by the minimum rank type parameters. We also give a method to calculate the $L$-Grundy domination number by the Grundy total domination number, giving some linear algebra bounds for the $L$-Grundy domination number.

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arXiv2016

Generalizations of the Strong Arnold Property and the minimum number of distinct eigenvalues of a graph

For a given graph G and an associated class of real symmetric matrices whose off-diagonal entries are governed by the adjacencies in G, the collection of all possible spectra for such matrices is considered. Building on the pioneering work of Colin de Verdiere in connection with the Strong Arnold Property, two extensions are devised that target a better understanding of all possible spectra and their associated multiplicities. These new properties are referred to as the Strong Spectral Property and the Strong Multiplicity Property. Finally, these ideas are applied to the minimum number of distinct eigenvalues associated with G, denoted by q(G). The graphs for which q(G) is at least the number of vertices of G less one are characterized.

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arXiv2016

Note on von Neumann and Rényi entropies of a Graph

We conjecture that all connected graphs of order $n$ have von Neumann entropy at least as great as the star $K_{1,n-1}$ and prove this for almost all graphs of order $n$. We show that connected graphs of order $n$ have Rényi 2-entropy at least as great as $K_{1,n-1}$ and for $α>1$, $K_n$ maximizes Rényi $α$-entropy over graphs of order $n$. We show that adding an edge to a graph can lower its von Neumann entropy.

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arXiv2016

Multi-part Nordhaus-Gaddum type problems for tree-width, Colin de Verdière type parameters, and Hadwiger number

A traditional Nordhaus-Gaddum problem for a graph parameter $β$ is to find a (tight) upper or lower bound on the sum or product of $β(G)$ and $β(\bar{G})$ (where $\bar{G}$ denotes the complement of $G$). An $r$-decomposition $G_1,\dots,G_r$ of the complete graph $K_n$ is a partition of the edges of $K_n$ among $r$ spanning subgraphs $G_1,\dots,G_r$. A traditional Nordhaus-Gaddum problem can be viewed as the special case for $r=2$ of a more general $r$-part sum or product Nordhaus-Gaddum type problem. We determine the values of the $r$-part sum and product upper bounds asymptotically as $n$ goes to infinity for the parameters tree-width and its variants largeur d'arborescence, path-width, and proper path-width. We also establish ranges for the lower bounds for these parameters, and ranges for the upper and lower bounds of the $r$-part Nordhaus-Gaddum type problems for the parameters Hadwiger number, the Colin de Verdière number $μ$ that is used to characterize planarity, and its variants $ν$ and $ξ$.

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arXiv2016

Using a new zero forcing process to guarantee the Strong Arnold Property

The maximum nullity $M(G)$ and the Colin de Verdière type parameter $ξ(G)$ both consider the largest possible nullity over matrices in $\mathcal{S}(G)$, which is the family of real symmetric matrices whose $i,j$-entry, $i\neq j$, is nonzero if $i$ is adjacent to $j$, and zero otherwise; however, $ξ(G)$ restricts to those matrices $A$ in $\mathcal{S}(G)$ with the Strong Arnold Property, which means $X=O$ is the only symmetric matrix that satisfies $A\circ X=O$, $I\circ X=O$, and $AX=O$. This paper introduces zero forcing parameters $Z_{\mathrm{SAP}}(G)$ and $Z_{\mathrm{vc}}(G)$, and proves that $Z_{\mathrm{SAP}}(G)=0$ implies every matrix $A\in \mathcal{S}(G)$ has the Strong Arnold Property and that the inequality $M(G)-Z_{\mathrm{vc}}(G)\leq ξ(G)$ holds for every graph $G$. Finally, the values of $ξ(G)$ are computed for all graphs up to $7$ vertices, establishing $ξ(G)=\lfloor Z\rfloor(G)$ for these graphs.

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arXiv2015

On the distance spectra of graphs

The distance matrix of a graph $G$ is the matrix containing the pairwise distances between vertices. The distance eigenvalues of $G$ are the eigenvalues of its distance matrix and they form the distance spectrum of $G$. We determine the distance spectra of halved cubes, double odd graphs, and Doob graphs, completing the determination of distance spectra of distance regular graphs having exactly one positive distance eigenvalue. We characterize strongly regular graphs having more positive than negative distance eigenvalues. We give examples of graphs with few distinct distance eigenvalues but lacking regularity properties. We also determine the determinant and inertia of the distance matrices of lollipop and barbell graphs.

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