scholarly journals The m-Path Cover Polynomial of a Graph and a Model for General Coefficient Linear Recurrences

2014 ◽  
Vol 2014 ◽  
pp. 1-13
Author(s):  
John P. McSorley ◽  
Philip Feinsilver
Keyword(s):  
General Term ◽  
Simple Graph ◽  
Time Dependent ◽  
Disjoint Paths ◽  
Third Way ◽  
Linear Recurrences ◽  
Path Cover ◽  
The Matrix ◽  
Cover Polynomial ◽  
Vertex Disjoint

An m-path cover Γ={Pℓ1,Pℓ2,…,Pℓr} of a simple graph G is a set of vertex disjoint paths of G, each with ℓk≤m vertices, that span G. With every Pℓ we associate a weight, ω(Pℓ), and define the weight of Γ to be ω(Γ)=∏k=1r‍ω(Pℓk). The m-path cover polynomial of G is then defined as ℙm(G)=∑Γ‍ω(Γ), where the sum is taken over all m-path covers Γ of G. This polynomial is a specialization of the path-cover polynomial of Farrell. We consider the m-path cover polynomial of a weighted path P(m-1,n) and find the (m+1)-term recurrence that it satisfies. The matrix form of this recurrence yields a formula equating the trace of the recurrence matrix with the m-path cover polynomial of a suitably weighted cycle C(n). A directed graph, T(m), the edge-weighted m-trellis, is introduced and so a third way to generate the solutions to the above (m+1)-term recurrence is presented. We also give a model for general-term linear recurrences and time-dependent Markov chains.

Partitioning a graph into vertex-disjoint paths

2005 ◽  
Vol 42 (3) ◽  
pp. 277-294
Author(s):  
Jianping Li ◽  
George Steiner
Keyword(s):  
Positive Integer ◽  
Sufficient Conditions ◽  
Simple Graph ◽  
Disjoint Paths ◽  
Extremal Graphs ◽  
Hamilton Path ◽  
Vertex Disjoint ◽  
Positive Integers

Let G=(V,E) be a simple graph of order n. We consider the problem of partitioning G into vertex-disjoint paths. We obtain the following new results: (i) For any positive integer k, if dG(x)+dG(y) = n-k-1 for every pair x, y of nonadjacent vertices in  G, then G can be partitioned into k vertex-disjoint paths, unless G belongs to certain classes of extremal graphs which we characterize; (ii) For the case k=2, we strengthen our result by showing that for any two positive integers p1 and p2 satisfying n= p1+ p2,if dG(x)+dG(y) = n-3  for every pair x, y of nonadjacent vertices in G and G does not belong to classes of exceptional graphs we define, then G can be partitioned into two vertex-disjoint paths  P1 and P2 of order p1 and p2,  respectively. These results are generalizations of some classical results of Dirac and Ore, and also lead to new sufficient conditions for the existence of a Hamilton path in a graph.

Unpaired Many-to-Many Disjoint Path Cover of Balanced Hypercubes

2021 ◽  
pp. 1-14
Author(s):  
Huazhong Lü ◽  
Tingzeng Wu
Keyword(s):  
Network Topology ◽  
Upper Bound ◽  
Interconnection Network ◽  
Disjoint Paths ◽  
Disjoint Path ◽  
Path Cover ◽  
Balanced Hypercube ◽  
Vertex Disjoint ◽  
Appl Math ◽  
Source And Sink

A many-to-many [Formula: see text]-disjoint path cover ([Formula: see text]-DPC) of a graph [Formula: see text] is a set of [Formula: see text] vertex-disjoint paths joining [Formula: see text] distinct pairs of source and sink in which each vertex of [Formula: see text] is contained exactly once in a path. The balanced hypercube [Formula: see text], a variant of the hypercube, was introduced as a desired interconnection network topology. Let [Formula: see text] and [Formula: see text] be any two sets of vertices in different partite sets of [Formula: see text] ([Formula: see text]). Cheng et al. in [Appl. Math. Comput. 242 (2014) 127–142] proved that there exists paired many-to-many 2-disjoint path cover of [Formula: see text] when [Formula: see text]. In this paper, we prove that there exists unpaired many-to-many [Formula: see text]-disjoint path cover of [Formula: see text] ([Formula: see text]) from [Formula: see text] to [Formula: see text], which has improved some known results. The upper bound [Formula: see text] is best possible in terms of the number of disjoint paths in unpaired many-to-many [Formula: see text]-DPC of [Formula: see text].

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.

Unpaired Many-to-Many Disjoint Path Covers on Bipartite k-Ary n-Cube Networks with Faulty Elements

2020 ◽  
Vol 31 (03) ◽  
pp. 371-383
Author(s):  
Jing Li ◽  
Chris Melekian ◽  
Shurong Zuo ◽  
Eddie Cheng
Keyword(s):  
Distributed Systems ◽  
Interconnection Networks ◽  
Disjoint Paths ◽  
Equal Size ◽  
Disjoint Path ◽  
Path Cover ◽  
The Many ◽  
Vertex Disjoint ◽  
Vertex Sets ◽  
Different Parts

The [Formula: see text]-ary [Formula: see text]-cube network is known as one of the most attractive interconnection networks for parallel and distributed systems. A many-to-many [Formula: see text]-disjoint path cover ([Formula: see text]-DPC for short) of a graph is a set of [Formula: see text] vertex-disjoint paths joining two disjoint vertex sets [Formula: see text] and [Formula: see text] of equal size [Formula: see text] that altogether cover every vertex of the graph. The many-to-many [Formula: see text]-DPC is classified as paired if each source in [Formula: see text] is further required to be paired with a specific sink in [Formula: see text], or unpaired otherwise. In this paper, we consider the unpaired many-to-many [Formula: see text]-DPC problem of faulty bipartite [Formula: see text]-ary [Formula: see text]-cube networks [Formula: see text], where the sets [Formula: see text] and [Formula: see text] are chosen in different parts of the bipartition. We show that, every bipartite [Formula: see text], under the condition that [Formula: see text] or less faulty edges are removed, has an unpaired many-to-many [Formula: see text]-DPC for any [Formula: see text] and [Formula: see text] subject to [Formula: see text]. The bound [Formula: see text] is tight here.

scholarly journals Expression of VEGF-A Signaling Pathway in Cartilage of ACLT-induced Osteoarthritis Mouse Model

2021 ◽  
Vol 16 (1) ◽  
Author(s):  
Jia-jia Qian ◽  
Qi Xu ◽  
Wei-min Xu ◽  
Ren Cai ◽  
Gui-cheng Huang
Keyword(s):  
Signaling Pathway ◽  
Cruciate Ligament ◽  
Time Dependent ◽  
Pathological Changes ◽  
The Matrix ◽  
Vegfr2 Signaling ◽  
Cox 2 ◽  
Anterior Cruciate ◽  
A Disintegrin And Metalloproteinase ◽  
Safranin O

Abstract Background Anterior cruciate ligament transection surgery (ACLT)-induced OA model was often used to investigate the molecular mechanism of knee osteoarthritis (KOA). Researches have shown that vascular endothelial growth factor (VEGF) played an important role in OA. The present study aimed to investigate the pathological changes after ACLT surgery and reveal the expression characteristics of the VEGF-A/VEGFR2 signaling pathway in this model. Methods Moderate KOA model was established by ACLT, and 1, 2, 4, 8, and 12 weeks after surgery, hematoxylin-eosin (HE) and Safranin-O(S-O) staining were used to detect the pathological changes in mouse knee cartilage, and the matrix biomarkers A Disintegrin and Metalloproteinase with Thrombospondin Motifs 5(ADAMTS5), Collagen II (COL-II) were detected using immunohistochemistry (IHC), CD31 was detected by immunofluorescence (IF) to show the vascular invasion in cartilage, and proteins expression of VEGF-A pathway were detected by Western blot (WB). Meanwhile, the inflammatory biomarkers cyclooxygenase-2 (COX-2) and inducible nitric oxide synthase (iNOS) in cartilage were detected by WB. Results ACLT surgery can lead to degeneration of cartilage in mice, and the characteristics of the lesion were time-dependent. The ADAMTS5-positive cells increased while COL-II decreased in OA cartilage with time, and new blood vessels labeled by CD31 can be seen from 1 week in OA cartilage, and increased in 8 and 12 weeks. The expression of VEGF-A, VEGFR2, COX-2, and iNOS were higher than control groups, which were basically consistent with the degree of osteoarthritis. Conclusions The degenerative degree of articular cartilage was time-dependent; angiogenesis and inflammation were important pathological changes of cartilage in KOA. The expression of the VEGF-A/VEGFR2 signaling pathway was basically correlated with the degree of KOA.

scholarly journals Cycle partitions of regular graphs

2020 ◽  
pp. 1-24
Author(s):  
Vytautas Gruslys ◽  
Shoham Letzter
Keyword(s):  
Regular Graph ◽  
Bipartite Graphs ◽  
Disjoint Paths ◽  
Regular Graphs ◽  
Simple Construction ◽  
Vertex Set ◽  
Almost All ◽  
Vertex Disjoint

Abstract Magnant and Martin conjectured that the vertex set of any d-regular graph G on n vertices can be partitioned into $n / (d+1)$ paths (there exists a simple construction showing that this bound would be best possible). We prove this conjecture when $d = \Omega(n)$ , improving a result of Han, who showed that in this range almost all vertices of G can be covered by $n / (d+1) + 1$ vertex-disjoint paths. In fact our proof gives a partition of V(G) into cycles. We also show that, if $d = \Omega(n)$ and G is bipartite, then V(G) can be partitioned into n/(2d) paths (this bound is tight for bipartite graphs).

scholarly journals A Numerical Approach for Evaluating the Time-Dependent Distribution of a Quasi Birth-Death Process

2021 ◽  
Author(s):  
Michel Mandjes ◽  
Birgit Sollie
Keyword(s):  
Continuous Time ◽  
Real Life ◽  
Numerical Approach ◽  
Time Dependent ◽  
Death Process ◽  
Continuous Time Markov Chain ◽  
Likelihood Estimator ◽  
Real Life Data ◽  
The Matrix ◽  
Birth Death

AbstractThis paper considers a continuous-time quasi birth-death (qbd) process, which informally can be seen as a birth-death process of which the parameters are modulated by an external continuous-time Markov chain. The aim is to numerically approximate the time-dependent distribution of the resulting bivariate Markov process in an accurate and efficient way. An approach based on the Erlangization principle is proposed and formally justified. Its performance is investigated and compared with two existing approaches: one based on numerical evaluation of the matrix exponential underlying the qbd process, and one based on the uniformization technique. It is shown that in many settings the approach based on Erlangization is faster than the other approaches, while still being highly accurate. In the last part of the paper, we demonstrate the use of the developed technique in the context of the evaluation of the likelihood pertaining to a time series, which can then be optimized over its parameters to obtain the maximum likelihood estimator. More specifically, through a series of examples with simulated and real-life data, we show how it can be deployed in model selection problems that involve the choice between a qbd and its non-modulated counterpart.

A UNIFIED FORMALISM OF THERMAL QUANTUM FIELD THEORY

1994 ◽  
Vol 09 (14) ◽  
pp. 2363-2409 ◽  
Author(s):  
H. CHU ◽  
H. UMEZAWA
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Vacuum Polarization ◽  
Time Dependent ◽  
Quantum Field ◽  
Bogoliubov Transformation ◽  
Transformation Matrices ◽  
Thermo Field Dynamics ◽  
The Matrix ◽  
Recent Developments

We present a comprehensive review of the most fundamental and practical aspects of thermo-field dynamics (TFD), including some of the most recent developments in the field. To make TFD fully consistent, some suitable changes in the structure of the thermal doublets and the Bogoliubov transformation matrices have been made. A close comparison between TFD and the Schwinger-Keldysh closed time path formalism (SKF) is presented. We find that TFD and SKF are in many ways the same in form; in particular, the two approaches are identical in stationary situations. However, TFD and SKF are quite different in time-dependent nonequilibrium situations. The main source of this difference is that the time evolution of the density matrix itself is ignored in SKF while in TFD it is replaced by a time-dependent Bogoliubov transformation. In this sense TFD is a better candidate for time-dependent quantum field theory. Even in equilibrium situations, TFD has some remarkable advantages over the Matsubara approach and SKF, the most notable being the Feynman diagram recipes, which we will present. We will show that the calculations of two-point functions are simplified, instead of being complicated, by the matrix nature of the formalism. We will present some explicit calculations using TFD, including space-time inhomogeneous situations and the vacuum polarization in equilibrium relativistic QED.

scholarly journals Mortality Risk and The Progression of The Electronic Frailty Index and Hospital Frailty Risk Score Over Time

2020 ◽  
Vol 5 (5) ◽  
Author(s):  
Joe Hollinghurst ◽  
Alan Watkins
Keyword(s):  
Risk Score ◽  
Frailty Index ◽  
Time Frame ◽  
Time Dependent ◽  
Cox Models ◽  
Hazard Ratios ◽  
Increased Risk ◽  
The Matrix ◽  
Time Dependent Covariates ◽  
Over Time

IntroductionThe electronic Frailty Index (eFI) and the Hospital Frailty Risk Score (HFRS) have been developed in primary and secondary care respectively. Objectives and ApproachOur objective was to investigate how frailty progresses over time, and to include the progression of frailty in a survival analysis.To do this, we performed a retrospective cohort study using linked data from the Secure Anonymised Information Linkage Databank, comprising 445,771 people aged 65-95 living in Wales (United Kingdom) on 1st January 2010. We calculated frailty, using both the eFI and HFRS, for individuals at quarterly intervals for 8 years with a total of 11,702,242 observations. ResultsWe created a transition matrix for frailty states determined by the eFI (states: fit, mild, moderate, severe) and HFRS (states: no score, low, intermediate, high), with death as an absorbing state. The matrix revealed that frailty progressed over time, but that on a quarterly basis it was most likely that an individual remained in the same state. We calculated Hazard Ratios (HRs) using time dependent Cox models for mortality, with adjustments for age, gender and deprivation. Independent eFI and HFRS models showed increased risk of mortality as frailty severity increased. A combined eFI and HFRS revealed the highest risk was primarily determined by the HFRS and revealed further subgroups of individuals at increased risk of an adverse outcome. For example, the HRs (95% Confidence Interval) for individuals with an eFI as fit, mild, moderate and severe with a high HFRS were 18.11 [17.25,19.02], 20.58 [19.93,21.24], 21.45 [20.85,22.07] and 23.04 [22.34,23.76] respectively with eFI fit and no HFRS score as the reference category. ConclusionFrailty was found to vary over time, with progression likely in the 8-year time-frame analysed. We refined HR estimates of the eFI and HFRS for mortality by including time dependent covariates.

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