Minimum and Maximum Resistance Given n resistors with resistance $R_1, R_2, \ldots, R_n$, and we can connect them with wires arbitrarily (this statement is not rigorous, but I think you understand what I mean), what is the minimum and maximum effective resistance? 
It's folklore that the minimum (which is $R_1\|R_2\|\cdots\|R_n$) is got when the resistors are connected in parallel and the maximum (which is $R_1+R_2+\cdots+R_n$) is got when the resistors are connected in series, but I do not see a way to prove this. I searched and only found that this result is used without proof. 
 A: There is a well-known theorem about electrical networks called Rayleigh's Monotonicity Theorem:
Theorem. Increasing the resistance of one resistor in a network cannot decrease the effective resistance of the network.
See this paper for a combinatorial proof of this theorem.  This has some basic corollaries:
Corollary. Removing a resistor from a network cannot decrease the resistance.
Proof: Removing a resistor is equivalent to increasing the resistance of the resistor to infinity. $\square$
Corollary. Replacing a resistor by a short cannot increase the resistance.
Proof: Replacing a resistor by a short is equivalent to decreasing the resistance of the resistor to zero. $\square$
We can use these corollaries to prove the minimum and maximum results you mentioned.
The Maximum Result
Given a network with resistors of resistances $R_1,\ldots,R_n$, choose a path from the initial vertex to the terminal vertex, and remove all of the resistors not on the path.  This does not decrease the resistance, and the resulting network is a simple series network involving some of the initial resistors.  In particular, the resulting network has resistance at most $R_1+\cdots + R_n$.
The Minimum Result
Given a network with resistors of resistances $R_1,\ldots,R_n$, choose a minimal cut set, i.e. a minimal set of resistors whose removal disconnects the initial vertex from the terminal vertex.  Replace all resistors that are not in this cut set by shorts.  This does not increase the resistance, and the result network is a simple parallel network involving some of the initial resistors.  In particular, the resulting network has resistance at least $R_1 \parallel \cdots \parallel R_n$
