# What is Vertex: Definition and 136 Discussions

In the mathematical discipline of graph theory, a vertex cover (sometimes node cover) of a graph is a set of vertices that includes at least one endpoint of every edge of the graph.
The problem of finding a minimum vertex cover is a classical optimization problem in computer science and is a typical example of an NP-hard optimization problem that has an approximation algorithm. Its decision version, the vertex cover problem, was one of Karp's 21 NP-complete problems and is therefore a classical NP-complete problem in computational complexity theory. Furthermore, the vertex cover problem is fixed-parameter tractable and a central problem in parameterized complexity theory.
The minimum vertex cover problem can be formulated as a half-integral linear program whose dual linear program is the maximum matching problem.
Vertex cover problems have been generalized to hypergraphs, see Vertex cover in hypergraphs.

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2. ### I Finding vertex of a 3D Triangle on a Plane

I came across the following problem and wondering how to solve it. There is a plane n1x + n2y + n3z + n4 = 0 where n1, n2, n3, n4 are known. The triangle is in this plane. We already know the two vertices P1(x1, y1, z1), P2(x2, y2, z2) of the triangle. Now we have to find the third vertex P(x...
3. ### A Renormalization with hard cutoff of a loop diagram with single vertex

Trying to solve for the loop contribution when renormalizing a one loop ##\frac{\lambda}{4!}\phi^4## diagram with two external lines in ##d=4## dimensions. After writing down the Feynman rule I see that: $$\frac{(-i\lambda)}{2}\int d^4q \frac{i}{q^2-m_{\phi}^2+i\epsilon}$$ But I see no way to...
4. ### A The vertex factors in QCD penguin operators

Have a look at O5 & O6 in Eqtns(5.4) . Why is there a (V+A) ? (V+A) contains the projection operator which projects out the right Weyl from a Dirac spinor. As per the Feynman rules of electroweak theory, there is a (V-A) assigned to each (Dirac) spinor-W boson vertex because W only couple to...
5. ### I Are Different Charged Gluons Repulsive in the Gluon-Gluon Four-Point Vertex?

Hello, First,i am an absolute beginner in QCD. As i have seen there are six different Feynmangraphs for the Four-Point vertex. Swordfish,Box,... I am interested in the first of the six where the two gluons met in a point and goes away. My question now is. If for this case two different...
6. ### I Causality for internel vertex in Feynaman diagrams

Eq 4.44 in Peskin and Schroeder. My question is: Does causality imply that time coordinates of z (internal vertex over which we doing the integration) should lie between the time coordinates of field phi(x) and phi(y)?
7. ### Cube with charges at each vertex

For convenience, I take the center of the upper face. The charges at the top cancel each other's effects, and those at the bottom cancel each other's horizontal effects, so I get$$E=4k\frac{q}{r^2}\sin\theta$$I have found that ##\theta=\arcsin\left(\frac{s}{r}\right)=\arcsin{\sqrt\frac{2}{3}}##...
8. ### A Two loop Feynman diagram with quartic vertex

I am trying to calculate the effective potential of two D0 branes scattering in Matrix theory and verify the coefficients in this paper: K. Becker and M. Becker, "A two-loop test of M(atrix) theory", Nucl. Phys. B 506 (1997) 48-60, arXiv:hep-th/9705091. The fields are expanded about a constant...
9. ### I No "i" in the Triple gluon vertex Feynman Rule

This is probably a simple question but puzzled me a little when trying to explain something to somebody. In all resources online and in most books I've seen, the triple gluon vertex has no overall 'i' factor while e.g. the four gluon vertex always does. The photon and gluon propagators as well...
10. ### Parabola: Vertex =(?,0) and know 2 arbitrary points. How solve x?

Summary:: I have a upwards opening parabola where I know the Y vertex point = 0. I also know 2 arbitrary points separated by a step size on the parabola. How can I solve for x at the vertex point? X=horizontal plane Y=vertical plane I have a upwards opening parabola where I know the Y vertex...
11. ### I Vertex function, quantum action

I am looking at Srednicki ch 64 , how does equation 64.1 follow from 64.3 as stated. Explicitly in QED how does ## u_{s'}(p')V^{u}(p',p)u_{s}(p)=e\bar{u'}(F_{1}(q^{2})\gamma ^{u}-\frac{i}{m}F_{2}(q^{2})S^{uv}q_{v})u ## follow from the quantum action ## \Gamma =\int d^{4}x(eF_{1}\bar{\varphi...
12. ### A Derivation of the Yang-Mills 3 gauge boson vertex

Hello everyone, I am stuck in the derivation of the three gauge-boson-vertex in Yang-Mills theories. The relevant interaction term in the Lagrangian is$$\mathcal{L}_{YM} \supset g \,f^{ijk}A_{\mu}{}^{(j)} A_{\nu}{}^{(k)} \partial^{\mu} A^{\nu}{}^{(i)}$$ I have rewritten this term using...
13. ### Three gauge boson Yang-Mills vertex

Hello everyone, I am stuck in deriving the three gauge-boson-vertex in Yang-Mills theories. The relevant interaction term in the Lagrangian is $$\mathcal{L}_{YM} \supset g \,f^{ijk}A_{\mu}{}^{(j)} A_{\nu}{}^{(k)} \partial^{\mu} A^{\nu}{}^{(i)}$$ I have rewritten this term using the...
14. ### MHB Find vertex, focus, and directrix of parabola: y^2+12y+16x+68=0

Find the vertex, focus, and directrix of the following parabola: y^2+12y+16x+68=0 The form we have been using is (y-k)^2=4p(x-h) Any explanation would help too, I'm really stuck on this one. Thank you!
15. ### A Electron Vertex Function in QED

In Peskin book they calculate the QED Vertex Function named Gamma which depend on two scalar functions called form factors F1(q^2), F2(q^2). they are calculating F1(q^2) and they do not really finish calculating for what I understand. They are just showing the Infrared divergent is canceled by...
16. ### I Some questions in QFT (EM vertex, Ward identity, etc.)

I am reading the A First Book of Quantum Field Theory. I have reached the chapter of renormalization, where the authors describe how the infinities of the self-energy diagrams can be corrected. They have also discussed later how the infrared and ultraviolet divergences are corrected. Just before...

50. ### Vertex corrections 1-loop order Yukawa theory

Consider the Yukawa theory ##\mathcal{L}_0 = \bar{\psi}_0(i\not \partial - m_0 - g\phi_0)\psi_0 + \frac{1}{2}(\partial \phi_0)^2 - \frac{1}{2}M_0^2 \phi_0^2 - \frac{1}{4!}\lambda_0 \phi_0^4## with cutoff ##\Lambda_0##; a lower cutoff ##\Lambda < \Lambda_0## is then introduced with an effective...