Simple & Connected Graph Definitions for Exam Tomorrow

terryfields
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just need 2 definitions without proof for an exam tomorrow, don't need to use them for anything just need to be able to quote them but can't find them anyway so if someone could helpfully write them down that would be great
1)simple graph
2)connected graph

cheers
 
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terryfields said:
just need 2 definitions without proof for an exam tomorrow, don't need to use them for anything just need to be able to quote them but can't find them anyway so if someone could helpfully write them down that would be great
1)simple graph
2)connected graph

cheers

Simple Graph - an undirected graph with no loops or multiple edges between the same two vertices.

Connected Graph - a graph in which all vertex points are joined by a path
 
Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...

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