# Laplacian Spectrum of Complete Graphs, Stars, and Rings

### Here's a little extra math to add to your understanding of graph theory.

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Join For Freea few examples help build intuition for what the eigenvalues of the graph laplacian tell us about a graph. the smallest eigenvalue is always zero (see explanation in footnote here ).

for a complete graph on
*
n
*
vertices, all the eigenvalues except the first equal
*
n
*
. the eigenvalues of the laplacian of a graph with
*
n
*
vertices are always less than or equal to
*
n
*
, this says the complete graph has the largest possible eigenvalue. this makes sense in light of the previous post:
adding edges increases the eigenvalues
. when you add as many edges as possible, you get the largest possible eigenvalues.

next we look at the case of a star with
*
n
*
-1 vertices connected to a central vertex. the smallest eigenvalue is zero, as always, and the largest is
*
n
*
. the ones in the middle are all 1. in particular, the
second eigenvalue
, the algebraic connectivity, is 1. it’s positive because the graph is connected, but it’s not large because the graph is not well connected: if you remove the central vertex it becomes completely disconnected.

finally, we look at a cyclical graph, a ring with
*
n
*
vertices. here the eigenvalues are 2 – 2 cos(2π
*
k
*
/
*
n
*
) where 0 ≤
*
k
*
≤
*
n/
*
2. in particular, smallest non-zero eigenvalue is 2 – 2 cos(2π/
*
n
*
) and so as
*
n
*
increases, the algebraic connectivity approaches zero. this is curious since the topology doesn’t change at all. but from a graph theory perspective, a big ring is less connected than a small ring. in a big ring, each vertex is connected to a small proportion of the vertices, and the average distance between vertices is larger.

Published at DZone with permission of John Cook, DZone MVB. See the original article here.

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