scholarly journals Relationship between adjacency and distance matrix of graph of diameter two

2021 ◽  
Vol 5 (2) ◽  
pp. 63
Author(s):  
Siti L. Chasanah ◽  
Elvi Khairunnisa ◽  
Muhammad Yusuf ◽  
Kiki A. Sugeng
Keyword(s):  
Upper Bound ◽  
Adjacency Matrix ◽  
Symmetric Matrix ◽  
Distance Matrix ◽  
Distance Matrices ◽  
The Relationship

The relationship among every pair of vertices in a graph can be represented as a matrix, such as in adjacency matrix and distance matrix. Both adjacency and distance matrices have the same property. Adjacency and distance matrices are both symmetric matrix with diagonals entries equals to 0.  In this paper, we discuss relationships between adjacency matrix and distance matrix of a graph of diameter two, which is <em>D=2(J-I)-A</em>. From this relationship, we  also determine the value of the determinant matrix <em>A+D</em> and the upper bound of determinant of matrix <em>D</em>.

scholarly journals The normalized distance Laplacian

2021 ◽  
Vol 9 (1) ◽  
pp. 1-18
Author(s):  
Carolyn Reinhart
Keyword(s):  
Spectral Radius ◽  
Characteristic Polynomial ◽  
Adjacency Matrix ◽  
Connected Graph ◽  
Distance Matrix ◽  
Laplacian Matrix ◽  
Diagonal Matrix ◽  
The Matrix

Abstract The distance matrix 𝒟(G) of a connected graph G is the matrix containing the pairwise distances between vertices. The transmission of a vertex vi in G is the sum of the distances from vi to all other vertices and T(G) is the diagonal matrix of transmissions of the vertices of the graph. The normalized distance Laplacian, 𝒟𝒧(G) = I−T(G)−1/2 𝒟(G)T(G)−1/2, is introduced. This is analogous to the normalized Laplacian matrix, 𝒧(G) = I − D(G)−1/2 A(G)D(G)−1/2, where D(G) is the diagonal matrix of degrees of the vertices of the graph and A(G) is the adjacency matrix. Bounds on the spectral radius of 𝒟 𝒧 and connections with the normalized Laplacian matrix are presented. Twin vertices are used to determine eigenvalues of the normalized distance Laplacian. The distance generalized characteristic polynomial is defined and its properties established. Finally, 𝒟𝒧-cospectrality and lack thereof are determined for all graphs on 10 and fewer vertices, providing evidence that the normalized distance Laplacian has fewer cospectral pairs than other matrices.

scholarly journals New Bounds for the α-Indices of Graphs

Mathematics ◽  
2020 ◽  
Vol 8 (10) ◽  
pp. 1668
Author(s):  
Eber Lenes ◽  
Exequiel Mallea-Zepeda ◽  
Jonnathan Rodríguez
Keyword(s):  
Lower Bound ◽  
Spectral Radius ◽  
Upper Bound ◽  
Adjacency Matrix ◽  
Independence Number ◽  
Minimal Degree ◽  
Diagonal Matrix ◽  
The Matrix

Let G be a graph, for any real 0≤α≤1, Nikiforov defines the matrix Aα(G) as Aα(G)=αD(G)+(1−α)A(G), where A(G) and D(G) are the adjacency matrix and diagonal matrix of degrees of the vertices of G. This paper presents some extremal results about the spectral radius ρα(G) of the matrix Aα(G). In particular, we give a lower bound on the spectral radius ρα(G) in terms of order and independence number. In addition, we obtain an upper bound for the spectral radius ρα(G) in terms of order and minimal degree. Furthermore, for n>l>0 and 1≤p≤⌊n−l2⌋, let Gp≅Kl∨(Kp∪Kn−p−l) be the graph obtained from the graphs Kl and Kp∪Kn−p−l and edges connecting each vertex of Kl with every vertex of Kp∪Kn−p−l. We prove that ρα(Gp+1)<ρα(Gp) for 1≤p≤⌊n−l2⌋−1.

A DIMENSIONLESS FIT MEASURE FOR PHYLOGENETIC DISTANCE TREES

2005 ◽  
Vol 03 (06) ◽  
pp. 1429-1440 ◽  
Author(s):  
MANUEL GIL ◽  
CHRISTOPHE DESSIMOZ ◽  
GASTON H. GONNET
Keyword(s):  
Phylogenetic Trees ◽  
Distance Matrix ◽  
Phylogenetic Distance ◽  
Linear Transformations ◽  
Relative Measure ◽  
Absolute Measure ◽  
Distance Matrices ◽  
Fit Index

We present a dimensionless fit index for phylogenetic trees that have been constructed from distance matrices. It is designed to measure the quality of the fit of the data to a tree in absolute terms, independent of linear transformations on the distance matrix. The index can be used as an absolute measure to evaluate how well a set of data fits to a tree, or as a relative measure to compare different methods that are expected to produce the same tree. The usefulness of the index is demonstrated in three examples.

Nonlinear Correlation Analysis of Time Series Based on Complex Network Similarity

2020 ◽  
Vol 30 (15) ◽  
pp. 2050225
Author(s):  
Chun-Xiao Nie
Keyword(s):  
Time Series ◽  
Correlation Analysis ◽  
Nearest Neighbor ◽  
Distance Matrix ◽  
Nonlinear Relationship ◽  
Nonlinear Correlation ◽  
Network Similarity ◽  
Level Of Significance ◽  
The Relationship ◽  
Analysis Of Time Series

Characterizing the relationship between time series is an important issue in many fields, in particular, in many cases there is a nonlinear correlation between series. This paper provides a new method to study the relationship between time series using the perspective of complex networks. This method converts a time series into a distance matrix and constructs a sequence of nearest neighbor networks, so that the nonlinear relationship between time series is expressed as similarity between networks. In addition, based on the surrogate series, we applied [Formula: see text]-score to characterize the level of significance and analyzed some benchmark models. We not only use the artificial dataset and the real dataset to verify the effectiveness of the proposed method, but also analyze its robustness, which provides an alternative method for detecting nonlinear relationships.

Concentration-QTc (C-QTc) Analysis of the MDM2 Inhibitor KRT-232 (formerly AMG 232) in Subjects with Advanced Solid Tumors, Multiple Myeloma, or Acute Myeloid Leukemia

Blood ◽  
2019 ◽  
Vol 134 (Supplement_1) ◽  
pp. 5768-5768
Author(s):  
Adekemi Taylor ◽  
Martine Allard ◽  
Cecile Kresja ◽  
Dana Lee ◽  
Greg Slatter
Keyword(s):  
Multiple Myeloma ◽  
Steady State ◽  
Solid Tumors ◽  
Upper Bound ◽  
Myeloid Leukemia ◽  
Employment Equity ◽  
Equity Ownership ◽  
The Mean ◽  
The Relationship ◽  
Mdm2 Inhibitor

Introduction: KRT-232 is a potent and selective, targeted small molecule inhibitor of human mouse double minute 2 (MDM2) homolog interactions with tumor protein 53 (p53). MDM2 prevents p53 activation and reduces p53-mediated transcription and cell cycle control. KRT-232 is under development by Kartos Therapeutics for treatment of myelofibrosis, polycythemia vera, acute myeloid leukemia (AML) and Merkel cell carcinoma (see NCT03662126, NCT03669965, NCT03787602). The KRT-232 no effect-level for in vitro inhibition of hERG function (10 μM) was approximately 147- and 73-fold greater than KRT-232 unbound Cmax concentrations for steady state doses of 240 mg and 480 mg, respectively, based on population pharmacokinetic (PK)-derived parameters for subjects with AML (Ma et al. submitted, ASH 2019). The primary objective of this analysis was to evaluate the relationship between KRT-232 plasma concentration and changes in heart rate-corrected QT interval duration (QTc) in oncology patients treated in Amgen studies 20120106 (Gluck et al. Invest New Drugs; in press, NCT01723020) and 20120234 (Erba et al. Blood Adv 2019; NCT02016729). Methods Study 20120106 was a 2-part Phase 1 dose-exploration and dose-expansion monotherapy study in advanced solid tumors or multiple myeloma. KRT-232 doses of 15 mg (n=3), 30 mg (n=3), 60 mg (n=4), 120 mg (n=7), 240 mg (n=76), 300 mg (n=4), 360 mg (n=4) and 480 mg (n=6) were administered daily (QD) for 7 days in 21-day cycles. Subjects received up to 31 cycles of treatment. Study 20120234 was a Phase 1b study evaluating KRT-232 alone and in combination with trametinib in relapsed/refractory AML. Subjects received the following KRT-232 doses: 60 mg (n=14; n=10 co-administered with 2 mg trametinib daily [excluded from C-QTc analysis]); n=4 as single agent), 90 mg (n=4), 180 mg (n=5), 240 mg (n=3), and 360 mg (n=10). Doses were administered QD for 7 days in 14-day cycles. Subjects received up to 46 cycles of treatment. In both studies, time-matched PK and ECG measurements were collected intensively during Cycle 1 and less frequently at other visits. Triplicate 12-lead ECG data (N=3) were read by a central laboratory. A linear mixed effects model using R (v 3.5.2) was used to analyze the relationship between KRT-232 plasma concentrations and the QT interval corrected using Fridericia's method (QTcF). Effects of baseline QTcF, study, sex and tumor type on C-QTc were investigated. The upper bound of 2-sided 90% CIs for the mean QTcF change from baseline (ΔQTcF) predicted at Cmax was compared to the 10 ms threshold of regulatory concern (FDA Guidance: E14(R3) 2017; Garnett et al. Pharmacokinet Pharmacodyn 2018). Results ECG and PK data for this analysis were available from 130 subjects. The final model was a linear mixed-effect model with parameters for intercept, KRT-232 concentration-ΔQTcF slope, and baseline QTcF effect on the intercept. Diagnostic plots indicated an adequate model fit. The final C-QTc model was used to predict mean ΔQTcF and associated 2-sided 90% CI mean steady-state KRT-232 Cmax at doses up to the maximum clinical dose of 480 mg QD, in subjects with AML or solid tumors. The mean and upper bound of the 90% CI of ΔQTcF were predicted not to exceed 10 ms at doses of up to 480 mg QD in subjects with AML, multiple myeloma or solid tumors. Mean (90% CI) predicted ΔQTcF values at 480 mg QD were 2.040 (0.486, 3.595) ms for subjects with solid tumors and 4.521 (2.348, 6.693) ms for subjects with AML (Figure A). The KRT-232 concentrations at which the upper bounds of 90% CI of mean ΔQTcF are predicted to reach 10 ms and 20 ms are 4298 ng/mL and 7821 ng/mL, respectively. These concentrations are 2.2- and 4-fold higher, respectively, than the predicted mean steady-state Cmax for 480-mg KRT-232 in subjects with solid tumors, and 1.4- and 2.5-fold higher, respectively, than the corresponding mean steady-state Cmax in subjects with AML. Conclusion Since the mean and upper bound of the 90% CI of mean ΔQTcF were predicted not to exceed 10 ms at KRT-232 doses of up to 480 mg QD in solid tumor or AML patients, KRT-232 should not result in clinically meaningful QT prolongation at the doses currently under investigation in Kartos clinical trials. Disclosures Taylor: Certara Strategic Consulting: Consultancy, Employment. Allard:Certara Strategic Consulting: Consultancy, Employment. Kresja:Kartos Therapeutics: Employment, Equity Ownership. Lee:Kartos Therapeutics: Employment, Equity Ownership. Slatter:Kartos Therapeutics: Employment, Equity Ownership. OffLabel Disclosure: KRT-232 (formerly AMG 232) is a small molecule MDM2 inhibitor

scholarly journals On Graphs with Cyclic Defect or Excess

10.37236/415 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Charles Delorme ◽  
Guillermo Pineda-Villavicencio
Keyword(s):  
Upper Bound ◽  
Adjacency Matrix ◽  
Minimum Degree ◽  
Integer Coefficients ◽  
Disjoint Cycles ◽  
The Matrix ◽  
Odd Degree ◽  
Unexplored Area ◽  
Trivial Graph ◽  
Vertex Disjoint

The Moore bound constitutes both an upper bound on the order of a graph of maximum degree $d$ and diameter $D=k$ and a lower bound on the order of a graph of minimum degree $d$ and odd girth $g=2k+1$. Graphs missing or exceeding the Moore bound by $\epsilon$ are called graphs with defect or excess $\epsilon$, respectively. While Moore graphs (graphs with $\epsilon=0$) and graphs with defect or excess 1 have been characterized almost completely, graphs with defect or excess 2 represent a wide unexplored area. Graphs with defect (excess) 2 satisfy the equation $G_{d,k}(A) = J_n + B$ ($G_{d,k}(A) = J_n - B$), where $A$ denotes the adjacency matrix of the graph in question, $n$ its order, $J_n$ the $n\times n$ matrix whose entries are all 1's, $B$ the adjacency matrix of a union of vertex-disjoint cycles, and $G_{d,k}(x)$ a polynomial with integer coefficients such that the matrix $G_{d,k}(A)$ gives the number of paths of length at most $k$ joining each pair of vertices in the graph. In particular, if $B$ is the adjacency matrix of a cycle of order $n$ we call the corresponding graphs graphs with cyclic defect or excess; these graphs are the subject of our attention in this paper. We prove the non-existence of infinitely many such graphs. As the highlight of the paper we provide the asymptotic upper bound of $O(\frac{64}3d^{3/2})$ for the number of graphs of odd degree $d\ge3$ and cyclic defect or excess. This bound is in fact quite generous, and as a way of illustration, we show the non-existence of some families of graphs of odd degree $d\ge3$ and cyclic defect or excess. Actually, we conjecture that, apart from the Möbius ladder on 8 vertices, no non-trivial graph of any degree $\ge 3$ and cyclic defect or excess exists.

scholarly journals Parental Resources and College Attendance: Evidence from Lottery Wins

2021 ◽  
Vol 111 (4) ◽  
pp. 1201-1240
Author(s):  
George Bulman ◽  
Robert Fairlie ◽  
Sarena Goodman ◽  
Adam Isen
Keyword(s):  
Financial Aid ◽  
Upper Bound ◽  
Financial Constraints ◽  
College Attendance ◽  
Financial Resources ◽  
Low Ses ◽  
Substantial Variation ◽  
College Costs ◽  
Us Children ◽  
The Relationship

We examine US children whose parents won the lottery to trace out the effect of financial resources on college attendance. The analysis leverages federal tax and financial aid records and substantial variation in win size and timing. While per-dollar effects are modest, the relationship is weakly concave, with a high upper bound for amounts greatly exceeding college costs. Effects are smaller among low-SES households, not sensitive to how early in adolescence the shock occurs, and not moderated by financial aid crowd-out. The results imply that households derive consumption value from college, and household financial constraints alone do not inhibit attendance. (JEL G51, I22, I23, I24, I26, I28, J24, J31)

scholarly journals Spectral Extrema for Graphs: The Zarankiewicz Problem

10.37236/212 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
László Babai ◽  
Barry Guiduli
Keyword(s):  
Bipartite Graph ◽  
Spectral Radius ◽  
Upper Bound ◽  
Adjacency Matrix ◽  
Complete Bipartite Graph ◽  
Average Degree ◽  
Largest Eigenvalue ◽  
Zarankiewicz Problem ◽  
The Largest Eigenvalue

Let $G$ be a graph on $n$ vertices with spectral radius $\lambda$ (this is the largest eigenvalue of the adjacency matrix of $G$). We show that if $G$ does not contain the complete bipartite graph $K_{t ,s}$ as a subgraph, where $2\le t \le s$, then $$\lambda \le \Big((s-1)^{1/t }+o(1)\Big)n^{1-1/t }$$ for fixed $t$ and $s$ while $n\to\infty$. Asymptotically, this bound matches the Kővári-Turán-Sós upper bound on the average degree of $G$ (the Zarankiewicz problem).

scholarly journals Skew Spectra of Oriented Graphs

10.37236/270 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Bryan Shader ◽  
Wasin So
Keyword(s):  
Directed Graph ◽  
Adjacency Matrix ◽  
Undirected Graph ◽  
Oriented Graph ◽  
Oriented Graphs ◽  
Underlying Graph ◽  
Adjacency Spectrum ◽  
The Relationship ◽  
Skew Adjacency Matrix

An oriented graph $G^{\sigma}$ is a simple undirected graph $G$ with an orientation $\sigma$, which assigns to each edge a direction so that $G^{\sigma}$ becomes a directed graph. $G$ is called the underlying graph of $G^{\sigma}$, and we denote by $Sp(G)$ the adjacency spectrum of $G$. Skew-adjacency matrix $S( G^{\sigma} )$ of $G^{\sigma}$ is introduced, and its spectrum $Sp_S( G^{\sigma} )$ is called the skew-spectrum of $G^{\sigma}$. The relationship between $Sp_S( G^{\sigma} )$ and $Sp(G)$ is studied. In particular, we prove that (i) $Sp_S( G^{\sigma} ) = {\bf i} Sp(G)$ for some orientation $\sigma$ if and only if $G$ is bipartite, (ii) $Sp_S(G^{\sigma}) = {\bf i} Sp(G)$ for any orientation $\sigma$ if and only if $G$ is a forest, where ${\bf i}=\sqrt{-1}$.

scholarly journals Object detection in images with a structural descriptor based on graphs

2018 ◽  
Vol 42 (2) ◽  
pp. 283-290 ◽  
Author(s):  
A. A. Zakharov ◽  
A. E. Barinov ◽  
A. L. Zhiznyakov ◽  
V. S. Titov
Keyword(s):  
Object Detection ◽  
Hausdorff Metric ◽  
Distance Matrix ◽  
Human Head ◽  
Image Features ◽  
Reference Object ◽  
Structural Descriptor ◽  
Centers Of Mass ◽  
Human Faces ◽  
The Relationship

We discuss the development of a structural descriptor for object detection in images. The descriptor is based on a graph, whose vertices are the centers of mass of segment features.  The embedding of the graph in a vector space is implemented using a Young-Householder decomposition and based on differential geometry. Compound curves are used to describe the relationship between the points. The image graph is described by a matrix of curvature parameters. The distance matrix for the graphs of the candidate object and the reference object is calculated using the Hausdorff metric. A multidimensional scaling method is used to represent the results. Images of test objects and images of human faces are used to study the developed approach. A comparison of the developed descriptor with the Viola-Jones method is performed when detecting a human head in the image. The advantage of the developed approach is the image rotational invariance in the plane while searching for objects. The descriptor can detect objects rotated in space by angles of up to 50 degrees. Using the mass centers of segments of features as the graph vertices makes the approach more robust to changes in image acquisition angles in comparison with the approach that uses image features as the graph vertices.

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