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{\bf Let $G$ be a connected graph.}

If $G =$ ~~~~~~~~~~~~ = ~~~~~ , then $\lambda(G) = $

In all other cases:

$\lambda(G)$  =  the size of a minimal edge cut

~~~~~~~ 	   =  the minimum number of edges one can
\u

~~~~~~~~~~  remove such that 
		 the resulting subgraph 
\u

~~~~~~~~~~
is disconnected.




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Thm 2.4: $\lambda(G) \leq min\{\delta(v) ~|~v \in V(G)\}$


Choose $w \in V(G)$ such that 

\rightline{$\delta(w) \leq \delta(v)$ for all $v \in V(G)$.  }



I.e., choose  $w \in V(G)$ such that 

\rightline{$\delta(w) = min\{\delta(v) ~|~v \in V(G)\}$}

Case 1:

\eject
Case 2:  Let $E^* = \{e ~|~ e$ is incident to $w\}$

\hskip 0.46in = $\{<w, v> ~|~ <w, v> ~~\in ~~E(G) \}$ 

\hskip 0.46in= $\{<w, v> ~|~  v ~\in~ N(w) \}$

\hskip 0.46in= the set of all edges having $w$ as one 

\rightline{of its vertices.}


Note $|E^*| = \delta(w)$.


 Note $(\{w\}, \emptyset)$ is a component of $G - E^*$. 

Thus $G - E^*$ is disconnected since 


\vfill

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$G = (V, E)$ is {\it $k$-edge-connected} if removing any set of $k-1$ edges in $G$ does not disconnect it and $|V| \geq 2$.

$k$ edge  connected implies $k-1$ edge connected if $k > 1$

%disconnected (more than one component).

connected = 1-edge connected 



$\lambda(G) = $ edge connectivity of $G$ 
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\rightline{= $max\{k ~|~ G ~k$-edge-connected $ \}$~~~~~~~~~~~~~~}
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\eject

Example of a 2-edge connected graph:
\v

Defn 2.13:  A {\bf Harary graph $H_{k, n}$} is a k-connected graph with $n$ vertices and a minimal number of edges.

Note $E(H_{k, n})= \lceil{kn \over 2}  \rceil$ 

\rightline{= the smallest natural number $\geq {kn \over 2}$.}


Thm 2.6:  If $k$ is even, then $G = (V, E)$ is a Harary graph where

 $V = \{0, 2, ...,n-1\}$ and 

$E = \bigcup\limits_{i \in V} \{<i, i + j> ~|~ j = 1, ..., {k \over 2}\}$

%\rightline{$ \cup \bigcup\limits_{i \in V} \{<i, i - j> ~|~ j = 1, ..., {k \over 2}\}$}

where addition is performed modulo $n$.
















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