Multivariate Bayesian formula I got there example  graphs bishop's PRML (8.2.1)
1. a <- c -> b
$$
p(a,b,c) = p(a|c)p(b|c)p(c)    --(1)\\
p(a,b) = \sum_c p(a|c)p(b|c)p(c)   --(2)
$$ 
Q1: Can I use a new graph to represent the p(a,b) in formula (2) ?
2.a->c->b
$$
p(a,b,c) = p(a)p(c|a)p(b|c)--(3)\\  
p(a,b) = p(a)\sum_c{p(c|a)p(b|c)} = p(a)p(b|a)  --(4)
$$
Q2: $$\sum_c{p(c|a)p(b|c)} = \sum_c{p(b,c|a)} ? why$$
 A: You have forgot to multiply with $p(a)$ to LHS side of your equation the summation actually results in $p(b/a)$. I would suggest you to recheck the equation in the book.
A: Intuitively, you may see the result of this formula by drawing the corresponding graph as follows:

The joint distribution represented by this graph is given by p(c|a)p(b|c)p(a). Now, summing out c, can be interpreted as removing the node labeled with c from the graph. So, the resulting graph is:

This new graph has the following joint distribution p(b|a)p(a).
Regarding a mathemtical view of Q2, we should note first that the book refers specifically at the graph a->c->b (Figure 8.17 in Bishop's Book) when talking about the equation in Q2.
p(b,c|a) = p(a, b, c)/p(a) "Definition of conditional probability"
p(b,c|a) = [p(b|c)p(c|a)p(a)]/p(a) "From the graph a->c->b we get the joint distribution"
p(b,c|a) = p(b|c)p(c|a) "Cancelling p(a) out"
Now, ∑cp(c|a)p(b|c)=∑cp(b,c|a)
∑cp(b,c|a) = p(b|a) "marginalization rule of probability"
I hope these steps make it clearer now.
