User:DVD206/An isospectral transformation of trivalent graphs

=An isospectral transformation of trivalent graphs=

Abstract. We describe an essentially isospectral transformation of trivalent graphs into four-valent graphs. The transformation is motivated by the construction of a medial graph for a triangulation of a surface. The transformation gives a family of counterexamples for the graph analog of the classic inverse problem "Can you hear the shape of a drum?".

A graph G(V,E) is a pair of sets, where E is a subset of pairs of distinct elements of the set V. The elements of V are called vertices, and the elements of E edges. For every edge (a,b) in E b is called a neighbor of a. The number of neighbors of a vertex is called its degree.

The Laplacian matrix L of a graph G(V,E) is defined in the following way:


 * $$L_{i,j}(G):=

\begin{cases} \deg(v_i) & \mbox{if}\ i = j \\ -1 & \mbox{if}\ i \neq j\ \mbox{and}\ (v_i v_j)\in E \\ 0 & \mbox{otherwise} \end{cases} $$

A graph is called trivalent if its every node has exactly three neighbors. If it is embedded into a surface then its dual graph is a triangulation of the surface.

If we define an oriented incidence matrix M with element Mev for edge e (connecting vertex i and j, with i > j) and vertex v given by
 * $$M_{ev} = \left\{ \begin{array}{rl}1, & \text{if}\,v=i\\-1, & \text{if}\,v=j\\0, & \text{otherwise},\end{array}\right.$$

then the Laplacian matrix L satisfies
 * $$L(G) = 6I-Q^{T}Q\,,$$

where $$Q^\text{T}$$ is the matrix transpose of Q.

The key observation of this paper is that
 * $$L(M(G))=6I-QQ^{T},$$

and, therefore, by the basic linear algebra the Laplacian spectra of the graphs G and M(G) are equal except the eigenvalue 6 w/some multiplicity.

The construction can be easily generalized to graphs of constant valency.

=References=
 * "Convex Polytopes". B. Grunbaum, Interscience, 1967
 * "Can one hear the shape of a drum?" M. Kac, American Mathematical Monthly 73 (4, part 2): 1–23, (1966)
 * "One cannot hear the shape of a drum". Carolyn Gordon, David L. Webb and Scott Wolpert, Bull. Amer. Math. Soc. 27 (1992), 134-138
 * "On 2D Inverse Problems". Wikibooks 2013