Bar Graphs: Practical Examples & Insights

Linear functions versus Logarithmic and Exponential functions
If a function is linear, I know that this should be true:
$f(x_2)-<!--\; -\; -->f(x_1)=f(\stackrel{¯<!--\; ¯\; -->}{x})$
Where $\stackrel{¯<!--\; ¯\; -->}{x}$ is the point exactly in between $x_1$ and $x_2$. Now, I know from looking at their graphs that the same shouldn't be true for logarithmic and exponential functions. That is, for logarithmic functions:
$f(x_2)-<!--\; -\; -->f(x_1)<f(\stackrel{¯<!--\; ¯\; -->}{x})$
And for exponential functions:
$f(x_2)-<!--\; -\; -->f(x_1)>f(\stackrel{¯<!--\; ¯\; -->}{x})$
I'm not a mathematician by formation, but I feel like it should be possible to prove this without having to look at their graph. Does anyone know how to do this?

How to demonstrate that $\mathrm{\chi <!--\; \chi \; -->}(\stackrel{¯<!--\; ¯\; -->}{G})=\mathrm{\omega <!--\; \omega \; -->}(\stackrel{¯<!--\; ¯\; -->}{G})$, G is a bipartite graph
If G is a bipartite graph then $\mathrm{\chi <!--\; \chi \; -->}(\stackrel{¯<!--\; ¯\; -->}{G})=\mathrm{\omega <!--\; \omega \; -->}(\stackrel{¯<!--\; ¯\; -->}{G})$?

chromatic number of a graph versus its complement
What can be said about the rate of growth of f(n), defined by
$f(n)=\underset{|V(G)|=n}{min}[\mathrm{\chi <!--\; \chi \; -->}(G)+\mathrm{\chi <!--\; \chi \; -->}(\stackrel{¯<!--\; ¯\; -->}{G})],$
where the minimum is taken over all graphs G on n vertices.
Two observations.
(1) Either G or $\stackrel{¯<!--\; ¯\; -->}{G}$ contains a clique on roughly logn vertices by Ramsey theory, so $f(n)\ge <!--\; \ge \; -->c_1\log <!--\; \; -->n$ for some constant $c_1>0$.
(2) If G=G(n,1/2) is a random graph, then $\mathrm{\chi <!--\; \chi \; -->}(G)\approx <!--\; \approx \; -->\mathrm{\chi <!--\; \chi \; -->}(\stackrel{¯<!--\; ¯\; -->}{G})\approx <!--\; \approx \; -->n/\log <!--\; \; -->n$ n/ log n almost surely, so we also have $f(n)\le <!--\; \le \; -->c_{2}n/\log <!--\; \; -->n$ n/log n for some constant $c_2>0$.
These bounds seem hopelessly far apart.
Can we improve on the bounds
$c_1\log <!--\; \; -->n\le <!--\; \le \; -->f(n)\le <!--\; \le \; -->c_{2}n/\log <!--\; \; -->n$
for all sufficiently large n?

Prove that a self-complementary graph has radius 2 and diameter 2 or 3.
I think that is one of the well-known properties of self-complementary graphs, but I am having some troubles trying to prove it. The facts that, if $G\cong <!--\; \cong \; -->\stackrel{¯<!--\; ¯\; -->}{G}$, then both graphs G and $\stackrel{¯<!--\; ¯\; -->}{G}$are connected, and that, for any graph, if $rad(G)\ge <!--\; \ge \; -->3$ then $rad(\stackrel{¯<!--\; ¯\; -->}{G})\le <!--\; \le \; -->2$, and that if $diam(G)\ge <!--\; \ge \; -->3$ then $diam(\stackrel{¯<!--\; ¯\; -->}{G})\le <!--\; \le \; -->3$, should make the proof quite easier, but I don't know how to develop it. Could you help me? Thanks in advance!

complete k-partite graphs
I am trying to solve the following problem:
Let G be a nonempty graph with the property that whenever $uv\notin <!--\; \notin \; -->E(G)$ and $vw\notin <!--\; \notin \; -->E(G)$, then $uw\notin <!--\; \notin \; -->E(G)$. Prove that G has this property if and only if G is a complete k-partite graph for some $k\ge <!--\; \ge \; -->2$. (Consider $\stackrel{¯<!--\; ¯\; -->}{G}$).
The converse is straightforward and is given by the definition of the complete k-partite graphs, however, the direct way is not trivial and I could not get it.

Prove $\mathrm{\chi <!--\; \chi \; -->}(G)\mathrm{\chi <!--\; \chi \; -->}(\stackrel{¯<!--\; ¯\; -->}{G})\ge <!--\; \ge \; -->n$ for chromatic number of graph and its complement
Let us denote by χ(G) the chromatic number, which is the smallest number of colours needed to colour the graph G with n vertices. Let $\stackrel{¯<!--\; ¯\; -->}{G}$ be the complement of G. Show that
$\mathrm{\chi <!--\; \chi \; -->}(G)+\mathrm{\chi <!--\; \chi \; -->}(\stackrel{¯<!--\; ¯\; -->}{G})\le <!--\; \le \; -->n+1$
$\mathrm{\chi <!--\; \chi \; -->}(G)\mathrm{\chi <!--\; \chi \; -->}(\stackrel{¯<!--\; ¯\; -->}{G})\ge <!--\; \ge \; -->n$
I was able to prove (a) using induction. Any hints on proving (b)?

Graph median, mode, range

The question is:
A scientist collects data. He determines both the mean of the data and the median of the data are equal to 7 and the data are symmetrical about this value. He starts to create a bar graph shown, but does not finish the graph.
If the range of the data is 8 and the maximum value of the data is 11, then how many data points fall above the value 7?
The answer is 14, but I don't know how this is the answer.

Prove that if diam(G)=3 for a regular graph G, then $diam(\stackrel{¯<!--\; ¯\; -->}{G})=2.$.
I know that without using regularity one can show that if $diam(G)\ge <!--\; \ge \; -->3,\text{then }diam(\stackrel{¯<!--\; ¯\; -->}{G})\le <!--\; \le \; -->3$. How do I use regularity to show that it must be exactly 2?

Boxplots and bar graphs
I'm studying for the GRE and came across several questions that I was unable to answer in a practice booklet, even after looking at the answer and trying to work backwards, and searching google and other sites for helpful hints. I think I am missing a fundamental understanding or useful heuristic for solving many of these problems; any advice would be greatly appreciated as my exam is Monday (Aug 1st).
1. Eight hundred insects were weighed, and the resulting measurements, in milligrams, are summarized in the boxplot below.
If the 80th percentile of the measurements is 130 milligrams, about how many measurements are between 126 milligrams and 130 milligrams?
I calculated the range (41), the quartiles(Q1=114, Q2=118, Q3=126), and the IQR (12), but I'm confused about the question. If the 80th percentile (so 80% of the measurements?) is at 130, then 640 are within this percentile. I'm not sure if this is true and even if it is, where to go from here. Each quartile is 25% of the data, correct? So from 126 to 146 must contain 25%, or 200 measurements? (1-.8)(200) = 40, but conceptually I'm lacking what that means.
2.This question refers to the following graph:

(a) In 2003 the family used a total of 49 percent of its gross annual income for two of the categories listed. What was the total amount of the family’s income used for those same categories in 2004 ?
My confusion lies with the fact that I can't seem to find any combination of two categories that add up to 49%. Plus, it seems that the total % expenditure in each year is 101%. The chart is not 100% accurately drawn, but even being very liberal in measuring there seems to be a discrepancy.