scholarly journals Maximum Multiplicity of Matching Polynomial Roots and Minimum Path Cover in General Graphs

10.37236/525 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Cheng Yeaw Ku ◽  
Kok Bin Wong
Keyword(s):  
Minimum Size ◽  
Polynomial Roots ◽  
Path Cover ◽  
Minimum Path ◽  
Matching Polynomial ◽  
Minimum Number ◽  
Vertex Set ◽  
The Difference ◽  
General Graphs ◽  
Maximum Multiplicity

Let $G$ be a graph. It is well known that the maximum multiplicity of a root of the matching polynomial $\mu(G,x)$ is at most the minimum number of vertex disjoint paths needed to cover the vertex set of $G$. Recently, a necessary and sufficient condition for which this bound is tight was found for trees. In this paper, a similar structural characterization is proved for any graph. To accomplish this, we extend the notion of a $(\theta,G)$-extremal path cover (where $\theta$ is a root of $\mu(G,x)$) which was first introduced for trees to general graphs. Our proof makes use of the analogue of the Gallai-Edmonds Structure Theorem for general root. By way of contrast, we also show that the difference between the minimum size of a path cover and the maximum multiplicity of matching polynomial roots can be arbitrarily large.

scholarly journals Maximum Multiplicity of a Root of the Matching Polynomial of a Tree and Minimum Path Cover

10.37236/170 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Cheng Yeaw Ku ◽  
K. B. Wong
Keyword(s):  
Disjoint Paths ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Path Cover ◽  
Minimum Path ◽  
Matching Polynomial ◽  
Minimum Number ◽  
Necessary And Sufficient ◽  
Vertex Disjoint ◽  
Maximum Multiplicity

We give a necessary and sufficient condition for the maximum multiplicity of a root of the matching polynomial of a tree to be equal to the minimum number of vertex disjoint paths needed to cover it.

The restrained k-rainbow reinforcement number of graphs

2020 ◽  
pp. 2150026
Author(s):  
Saeed Shaebani ◽  
Saeed Kosari ◽  
Leila Asgharsharghi
Keyword(s):  
Open Neighborhood ◽  
Minimum Weight ◽  
Domination Number ◽  
Simple Graph ◽  
The Family ◽  
Rainbow Domination ◽  
Minimum Number ◽  
Vertex Set ◽  
Number Formula ◽  
General Graphs

Let [Formula: see text] be a positive integer and [Formula: see text] be a simple graph. A restrained [Formula: see text]-rainbow dominating function (R[Formula: see text]RDF) of [Formula: see text] is a function [Formula: see text] from the vertex set [Formula: see text] to the family of all subsets of [Formula: see text], such that every vertex [Formula: see text] with [Formula: see text] satisfies both of the conditions [Formula: see text] and [Formula: see text] simultaneously, where [Formula: see text] denotes the open neighborhood of [Formula: see text]. The weight of an R[Formula: see text]RDF is the value [Formula: see text]. The restrained[Formula: see text]-rainbow domination number of [Formula: see text], denoted by [Formula: see text], is the minimum weight of an R[Formula: see text]RDF of [Formula: see text]. The restrained[Formula: see text]-rainbow reinforcement number [Formula: see text] of [Formula: see text], is defined to be the minimum number of edges that must be added to [Formula: see text] in order to decrease the restrained [Formula: see text]-rainbow domination number. In this paper, we determine the restrained [Formula: see text]-rainbow reinforcement number of some special classes of graphs. Also, we present some bounds on the restrained [Formula: see text]-rainbow reinforcement number of general graphs.

scholarly journals Intrinsic linking and knotting in tournaments

2019 ◽  
Vol 28 (12) ◽  
pp. 1950076
Author(s):  
Thomas Fleming ◽  
Joel Foisy
Keyword(s):  
Directed Graph ◽  
Upper Bound ◽  
Directed Edge ◽  
Undirected Graphs ◽  
Oriented Cycle ◽  
Minimum Number ◽  
The Difference

A directed graph [Formula: see text] is intrinsically linked if every embedding of that graph contains a nonsplit link [Formula: see text], where each component of [Formula: see text] is a consistently oriented cycle in [Formula: see text]. A tournament is a directed graph where each pair of vertices is connected by exactly one directed edge. We consider intrinsic linking and knotting in tournaments, and study the minimum number of vertices required for a tournament to have various intrinsic linking or knotting properties. We produce the following bounds: intrinsically linked ([Formula: see text]), intrinsically knotted ([Formula: see text]), intrinsically 3-linked ([Formula: see text]), intrinsically 4-linked ([Formula: see text]), intrinsically 5-linked ([Formula: see text]), intrinsically [Formula: see text]-linked ([Formula: see text]), intrinsically linked with knotted components ([Formula: see text]), and the disjoint linking property ([Formula: see text]). We also introduce the consistency gap, which measures the difference in the order of a graph required for intrinsic [Formula: see text]-linking in tournaments versus undirected graphs. We conjecture the consistency gap to be nondecreasing in [Formula: see text], and provide an upper bound at each [Formula: see text].

scholarly journals The Non-Commuting, Non-Generating Graph of a Nilpotent Group

10.37236/9802 ◽  
2021 ◽  
Vol 28 (1) ◽  
Author(s):  
Peter Cameron ◽  
Saul Freedman ◽  
Colva Roney-Dougal
Keyword(s):  
Maximal Subgroup ◽  
Nilpotent Group ◽  
Commuting Graph ◽  
Generating Graph ◽  
Central Elements ◽  
Vertex Set ◽  
Infinite Case ◽  
The Difference ◽  
The Relationship

For a nilpotent group $G$, let $\Xi(G)$ be the difference between the complement of the generating graph of $G$ and the commuting graph of $G$, with vertices corresponding to central elements of $G$ removed. That is, $\Xi(G)$ has vertex set $G \setminus Z(G)$, with two vertices adjacent if and only if they do not commute and do not generate $G$. Additionally, let $\Xi^+(G)$ be the subgraph of $\Xi(G)$ induced by its non-isolated vertices. We show that if $\Xi(G)$ has an edge, then $\Xi^+(G)$ is connected with diameter $2$ or $3$, with $\Xi(G) = \Xi^+(G)$ in the diameter $3$ case. In the infinite case, our results apply more generally, to any group with every maximal subgroup normal. When $G$ is finite, we explore the relationship between the structures of $G$ and $\Xi(G)$ in more detail.

scholarly journals On L h , k -Labeling Index of Inverse Graphs Associated with Finite Cyclic Groups

2021 ◽  
Vol 2021 ◽  
pp. 1-7
Author(s):  
K. Mageshwaran ◽  
G. Kalaimurugan ◽  
Bussakorn Hammachukiattikul ◽  
Vediyappan Govindan ◽  
Ismail Naci Cangul
Keyword(s):  
Cyclic Group ◽  
Upper Bound ◽  
Labeling Index ◽  
Positive Difference ◽  
Cyclic Groups ◽  
Finite Cyclic Group ◽  
Minimum Number ◽  
The Difference ◽  
Radio Labeling

An L h , k -labeling of a graph G = V , E is a function f : V ⟶ 0 , ∞ such that the positive difference between labels of the neighbouring vertices is at least h and the positive difference between the vertices separated by a distance 2 is at least k . The difference between the highest and lowest assigned values is the index of an L h , k -labeling. The minimum number for which the graph admits an L h , k -labeling is called the required possible index of L h , k -labeling of G , and it is denoted by λ k h G . In this paper, we obtain an upper bound for the index of the L h , k -labeling for an inverse graph associated with a finite cyclic group, and we also establish the fact that the upper bound is sharp. Finally, we investigate a relation between L h , k -labeling with radio labeling of an inverse graph associated with a finite cyclic group.

Finding Minimum Reaction Cuts of Metabolic Networks Under a Boolean Model Using Integer Programming and Feedback Vertex Sets

2012 ◽  
pp. 774-791
Author(s):  
Takeyuki Tamura ◽  
Kazuhiro Takemoto ◽  
Tatsuya Akutsu
Keyword(s):  
Integer Programming ◽  
Metabolic Network ◽  
Drug Targets ◽  
Metabolic Networks ◽  
Boolean Model ◽  
Pentose Phosphate ◽  
Minimum Size ◽  
E Coli ◽  
Potential Applications ◽  
Vertex Set

In this paper, the authors consider the problem of, given a metabolic network, a set of source compounds and a set of target compounds, finding a minimum size reaction cut, where a Boolean model is used as a model of metabolic networks. The problem has potential applications to measurement of structural robustness of metabolic networks and detection of drug targets. They develop an integer programming-based method for this optimization problem. In order to cope with cycles and reversible reactions, they further develop a novel integer programming (IP) formalization method using a feedback vertex set (FVS). When applied to an E. coli metabolic network consisting of Glycolysis/Glyconeogenesis, Citrate cycle and Pentose phosphate pathway obtained from KEGG database, the FVS-based method can find an optimal set of reactions to be inactivated much faster than a naive IP-based method and several times faster than a flux balance-based method. The authors also confirm that our proposed method works even for large networks and discuss the biological meaning of our results.

Identification of the Minimum Number of Measurements Required for Thermal Comfort Assessment in Large Workplaces

2019 ◽  
Vol 63 (7) ◽  
pp. 729-742
Author(s):  
Paolo Lenzuni ◽  
Pierangelo Tura ◽  
Pierfrancesco Cervino
Keyword(s):  
Thermal Comfort ◽  
Thermal Environment ◽  
Significant Loss ◽  
Spatial Inhomogeneity ◽  
Main Window ◽  
Room Size ◽  
Synthetic Index ◽  
Minimum Number ◽  
The Difference ◽  
A Minor

Abstract Optimization of resources is the key to improve our ability to perform multiple tasks with limited time and money. In the context of thermal comfort assessment, optimization becomes important in large rooms where tens of individuals perform similar tasks. This work focuses on the identification of the minimum number of measurement points that allows an accurate description of the thermal environment. Accuracy of description is assumed if no significant loss of information is associated to the transition from the ‘primary’ thermal map based on all available measurement points to a ‘secondary’ thermal map based on a reduced set of measurement points. The concept of ‘no significant loss’ is quantified by requiring that the difference in PMV (Predicted Mean Vote) between the two maps is kept <0.1 in the vast majority (95%) of points. PMV is a standardized synthetic index that is used worldwide for quantifying thermal comfort (ISO 7730, 2005) taking into account both environmental (thermo-hygrometric) and personal (activity, clothing) quantities. We show that the uncertainty induced by the degraded resolution of the thermal map has a limited impact on the overall uncertainty on PMV. Application of the method to a few test environments shows that the room size perpendicular to the main window and the windows orientation play the largest role in determining spatial inhomogeneity in thermal maps. A minor concurring factor is the room area.

Total k-rainbow reinforcement number in graphs

2020 ◽  
pp. 2050101
Author(s):  
L. Shahbazi ◽  
H. Abdollahzadeh Ahangar ◽  
R. Khoeilar ◽  
S. M. Sheikholeslami
Keyword(s):  
Open Neighborhood ◽  
Minimum Weight ◽  
Domination Number ◽  
The Family ◽  
Rainbow Domination ◽  
Minimum Number ◽  
Vertex Set ◽  
Isolated Vertex ◽  
Complete Bipartite Graphs ◽  
Dominating Function

Let [Formula: see text] be an integer, and let [Formula: see text] be a graph. A k-rainbow dominating function (or [Formula: see text]RDF) of [Formula: see text] is a function [Formula: see text] from the vertex set [Formula: see text] to the family of all subsets of [Formula: see text] such that for very [Formula: see text] with [Formula: see text], the condition [Formula: see text] is fulfilled, where [Formula: see text] is the open neighborhood of [Formula: see text]. The weight of a [Formula: see text]RDF [Formula: see text] of [Formula: see text] is the value [Formula: see text]. A k-rainbow dominating function [Formula: see text] in a graph with no isolated vertex is called a total k-rainbow dominating function if the subgraph of [Formula: see text] induced by the set [Formula: see text] has no isolated vertices. The total k-rainbow domination number of [Formula: see text], denoted by [Formula: see text], is the minimum weight of the total [Formula: see text]-rainbow dominating function on [Formula: see text]. The total k-rainbow reinforcement number of [Formula: see text], denoted by [Formula: see text], is the minimum number of edges that must be added to [Formula: see text] in order to decrease the total k-rainbow domination number. In this paper, we investigate the properties of total [Formula: see text]-rainbow reinforcement number in graphs. In particular, we present some sharp bounds for [Formula: see text] and we determine the total [Formula: see text]-rainbow reinforcement number of some classes of graphs including paths, cycles and complete bipartite graphs.

Strain Differences Occurring in an Experimental Study of Punishment

1965 ◽  
Vol 16 (2) ◽  
pp. 531-536 ◽  
Author(s):  
Morton H. Kleban
Keyword(s):  
Experimental Study ◽  
Empirical Evidence ◽  
Wistar Rats ◽  
Strain Differences ◽  
Training Procedure ◽  
Sprague Dawley ◽  
Sprague Dawley Rats ◽  
Y Maze ◽  
Minimum Number ◽  
The Difference

Forty-three Sprague-Dawley and 43 Wistar rats were given reward training for 40 trials in a Y-maze. On the next 20 trials, control groups were continued under the same training procedure, and 50% shock trials were introduced in the training of the remaining rats. For the extinction training, the reward was shifted to the opposite arm and 50% shock was continued for the no-delay and 30-sec. delay shock groups. The most significant results were that in the 30-sec. delay groups, the delay helped the Sprague-Dawley rats reverse in a minimum number of trials, whereas the Wistar rats showed strong indications of response stereotypy. The findings with respect to the Sprague-Dawley rats supported the empirical evidence on the effectiveness of delay in overcoming response persistence and the findings on the Wistar rats supported the empirical evidence on omission in punishment. The difference in response to punishment between the two albino strains emphasizes the need for experimental study of strain factors. Experiments should be repeated with several animal strains to remedy over-generalization from single strains and to help elaborate our understanding of the interaction present between punishment and strains.

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