New families of graphs determined by their generalized spectrum

Abstract

We construct infinite families of graphs that are determined by their generalized spectrum. This construction is based on new formulae for the determinant of the walk matrix of a graph. The graphs constructed here all satisfy a lower divisibility for the determinant of their walk matrix. 1 spectrum, or a DS graph for short. In [4,8,9] it is conjectured that almost all graphs are DS, more recent surveys can be found in [5].A variant of this problem concerns the generalized spectrum of G which is given by the pair spec(G), spec(G) . In this situation G is said to be determined by its generalized spectrum, or a DGS graph for short, if spec(H), spec(H) = spec(G), spec(G) implies that H is isomorphic to G. For the most recent results on DGS graphs see [13] and [20] in particular. To date only a few families of graphs are known to be DS or DGS. This includes some almost complete graphs [1], pineapple graphs [10] and kite graphs [11]. These are all known to be DS, and in particular DGS. In addition all rose graphs are determined by their Laplacian spectrum except for two specific examples, see [12].In this paper we construct infinite sequences of DGS graphs from certain small starter graphs. This construction involves the walk matrix of a graph and recent results in [20]. Let n be the number of vertices of G and let e be the column vector of height n with all entries equal to 1. Then the walk matrix of G is the n × n matrix W = e, Ae, A 2 e, · · · , A n−1 e

Generalized spectral characterizations of almost controllable graphs

“…The overall idea comes from [12] and [13], but difficulty inevitably arises due to the increase of the dimension of the nullspace N(W T (G)) by one. Before going into the details, we first give some structural properties on the two-dimensional nullspace N(W T (G)), which will results in an equivalent description of (5).…”

“…Using Theorem 3 as the new starting point, the full proof of Theorem 5 is given in Sec. 5. Some examples are given to illustrate Theorem 5 in the final section.…”

Unlocking the walk matrix of a graph

“…The matrix f j • f T j is the projection of K n onto the hyperplane with normal f j and so the sum on the left represents the eigenspace decomposition of ker(W T ). From (17) we have…”

“…A survey about walk matrices and the related topic of main eigenvalues and main eigenvectors can be found in [20]. More recently walk matrices have been studied in spectral graph theory [27][28][29] and in particular in connection with the question whether a graph is identified up to isomorphism by its spectrum [17].…”

Unlocking the walk matrix of a graph

“…Surveys about the related topics of main eigenvalues and main eigenvectors can be found in [16]. More recently walk matrices have been studied in spectral graph theory, see W. Wang [24,25,26], and in particular regarding the question whether a graph is identified up to isomorphism by its spectrum, see [13].…”