Spectra of Sums of Networks

What happens with the spectrum of the Laplacian L if we add some graphs or simplicial complexes? (I owe this question to An Huang). Here is an example, where we sum a circular graph G=C4 and a star graph H=S4. The Laplace spectrum of G is {4,2,2,0}, the spectrum of H is {5,1,1,1,0}. The spectrum of G+H is {9, 9, 9, 7, 7, 5, 5, 5, 0}.

The largest eigenvalue

In this post just three remarks about the spectrum of sums. The first one deals with the maximal eigenvalue. In some sense, when we look at sums of graphs, we are in an extreme situation, where the maximal eigenvalue is as large as it can be. This is a bit surprising as we would expect that in general the maximal eigenvalue is smaller. But as we see, if the maximal eigenvalue is smaller, then we have an additive prime graph.

If G=H+K then there is an eigenvalue |V(H)| + |V(K)| of the scalar Laplacian L. Consequently, any graph G for which there is no eigenvalue v_0(G) for L is an additive prime.

Proof: If m=|V(H)| and n=|V(K)|, then the vector which is constant n on H and constant m on K is an eigenvector to the Laplacian L with eigenvalue n+m. The eigenvector is perpendicular to the constant.

Here is a corollary, which can be seen as a result in the ring of graphs as
n G is Kn . G is a multiple of a graph.

The graph n G= G+G+ \cdots + G has the eigenvalue n |V(G)| with multiplicity at least n-1.

The multiplicity follows because we can write the sum in n-1 different ways as a sum of two graphs A+B. The proof above determine then the eigenvector.

The example K_n=K_1+K_1+ \dots K_1, where n appears with multiplicity n-1 is an extreme case. An other case is the (n-1)-dimensional cross polytop n P_2 = P_2 + \cdots +  P_2 which has the eigenvalue 2n with multiplicity n-1.

I’m confident that there are more relations in general between the spectral of H.K and the spectra of H and K.

The second eigenvalue

Graphs always have the constant eigenvector to the eigenvalue 0. The first interest is the second eigenvalue which has been studied much, both in graph theory as well as differential geometry. It is somehow the ground state as we don’t want to count the zero eigenstate as something interesting. What is interesting is the first eigenvalue gap, which in the case of graphs is measured by the smallest non-zero eigenvalue \lambda_2 of the Laplacian.

\lambda_2(G+H) = {\rm min}(|V(H)|,|V(K)|) + {\rm min}(\lambda_2(G),\lambda_2(H).

Proof: The Courant-Fischer formula tells \lambda_2 = {\rm inf}_{v \cdot 1 = 0} (v,Lv)/(v,v). Let v_2(H) be the eigenvector of L(H) with eigenvalue \lambda_2(H) and v_2(G) the eigenvector of L(H) with eigenvalue \lambda_2(G). Assume they are normalized. Extend v_2(G) and v_2(H) onto G+H by putting 0 on the other part. They still have norm 1 on the product space. They are also perpendicular to the constant vector. Now (v_2(H),L v_2(H))=\lambda_2(H)+m and (v_2(G),L v_2(G)) = \lambda_2(G) + n. This shows \lambda_2(G+H) \leq  {\rm min}(|V(H)|,|V(K)|) + {\rm min}(\lambda_2(G),\lambda_2(H) \; . On the other hand, since G+H can be obtained from the disjoint union of G and P_n (the graph without edges), by adding edges, we have \lambda_i(G) + n \leq \lambda_i(G+H). Similarly \lambda_i(H) + m \leq  \lambda_i(G+H).

Here we see as an example a sequence of graphs H,H+H,H+H+H, H+H+H+H with H=P2. These are 0,1,2,3 – dimensional spheres (cross polytopes) The eigenvalues are {0,0},{4,2,2,0},{6,6,4,4,4,0},{8,8,8,6,6,6,6,0}. Since H+H=C4 we can also write the last 3-sphere as C4+C4 = K2 . C4 = 4 P2.

The volume Laplacian

Here is an other spectral result. It deals with the highest form Laplacian. (See this document for some background.) Let D=d+d^* be the Dirac operator of a simplicial complex. The form Laplacian L=(d+d^*)^2 splits into blocks called form Laplacians L_k. The nullity of L_k is the k‘th Betti number b_k see. We have already seen that if we have two graphs G,H, then the volume of G+H is the product of the volumes of G and volumes of H. Lets write L_v(G) for L_{{\rm dim}(G)}(G) for the volume Laplacian, the Laplacian belonging to the largest dimension. Now if G has volume m and H has volume n then we can look at the volume eigenvalues \lambda_1, \dots, \lambda_n of L_v(G) and the volume eigenvalues \mu_1, \dots , \mu_m of L_v(H). What are the volume eigenvalues of H+G?

For two arbitrary networks, the volume eigenvalues of H+G are given by \lambda+\mu, where \lambda runs over the volume eigenvalues of H and \mu runs over the volume eigenvalues of G.

Proof: If U is the volume Laplacian of G and K is the volume Laplacian of H. Given U v = \lambda v and V w = \mu w. The volume Laplacian W of G+H is a (nm) \times (nm) matrix as the volumes of G and K have multiplied. Define on the facet xy of G+H joining the facets x and y of G and H the value f_{xy} = v_x w_y. Now W f = \lambda f + \mu f.

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