Symmetric matrices, signed graphs, and nodal domain theorems

Nodal domain theorems for $p$-Laplacians on signed graphs

“…In this paper, we systematically establish a nodal domain theory for p-Laplacians on signed graphs, which unifies the ideas and approaches from these recent works [23,30,38,49]. Based on our nodal domain estimates, we also obtain a higher order Cheeger inequality that relates the variational eigenvalues of p-Laplacians and Atay-Liu's multi-way Cheeger constants on signed graphs [6].…”

“…Therefore, it should be natural and useful to develop a general spectral theory that includes nodal domain theorems on signed graphs. Along this line, Ge and Liu [30] provided a definition of the strong and weak nodal domains on signed graphs, which is compatible with the classical one in [22] on graphs. They also obtained sharp estimates of the number of strong and weak nodal domains for generalized linear Laplacian on signed graphs.…”

“…We notice that estimates of strong nodal domains on signed graphs has been established in an earlier work of Mohammadian [43], see [30,Remark 3.12]. For more details and historical background of nodal domain theory, we refer the readers to [30]. We particularly mention that the results in Fiedler's classical 1975 paper [26] can be considered as nodal domain theorems on signed trees (see [30,Section 5]).…”

“…They also obtained sharp estimates of the number of strong and weak nodal domains for generalized linear Laplacian on signed graphs. We notice that estimates of strong nodal domains on signed graphs has been established in an earlier work of Mohammadian [43], see [30,Remark 3.12]. For more details and historical background of nodal domain theory, we refer the readers to [30].…”

“…For more details and historical background of nodal domain theory, we refer the readers to [30]. We particularly mention that the results in Fiedler's classical 1975 paper [26] can be considered as nodal domain theorems on signed trees (see [30,Section 5]). In 2013, Berkolaiko [8] and Colin de Verdière [18] computed the nodal count of edges on signed graphs by allowing the signs of each edge to become complex.…”