arXiv subjects

Takumi Gomyou

Publications and source records attributed to Takumi Gomyou.

3 recordsLinked to original sources

arXiv2026

Maximization of the first Laplace eigenvalue of a finite graph II

Given a length function on the set of edges of a finite graph, the corresponding Fujiwara Laplacian is defined. We consider a problem of maximizing the first nonzero eigenvalue of this graph Laplacian over all choices of edge-length function subject to a certain normalization. In this paper we prove that the supremum of the first nonzero eigenvalue is finite if and only if the graph is a tree. We also prove that the supremum of the first nonzero eigenvalue is nonincreasing under taking a subgraph.

math.CO↗

arXiv2021

Embedding and the rotational dimension of a graph containing a clique

The rotational dimension is a minor monotone graph invariant related to the dimension of an Euclidean space containing a spectral embedding corresponding to the first nonzero eigenvalue of the graph Laplacian, which is introduced by Göring, Helmberg and Wappler. In this paper, we study rotational dimensions of graphs which contain large complete graphs. The complete graph is characterized by its rotational dimension. It will be a obtained that a chordal graph may be made large while keeping the rotational dimension constant.

math.CO↗

arXiv2020

Optimal embedding and spectral gap of a finite graph

We introduce a new optimization problem regarding embeddings of a graph into a Euclidean space and discuss its relation to the two, mutually dual, optimizations problems introduced by Goering-Helmberg-Wappler. We prove that the Laplace eigenvalue maximization problem of Goering et al is also dual to our embedding optimization problem. We solve the optimization problems for generalized polygons and graphs isomorphic to the one-skeltons of regular and semi-regular polyhedra.

math.CO↗