Let G be a simple undirected graph on n vertices and let S(G) be the class of real symmetric n× n matrices whose nonzero off-diagonal entries correspond exactly to the edges of G. Given 2n − 1 real numbers λ1 ≥ μ1 ≥ λ2 ≥ μ2 ≥ · · · ≥ λn−1 ≥ μn−1 ≥ λn, and a vertex v of G, the question is addressed of whether or not there exists A ∈ S(G) with eigenvalues λ1, . . . , λn such that A(v) has eigenva...

Abstract We study the problem of determining maximum size a spherical two-distance set with two fixed angles (one acute and one obtuse) in high dimensions. Let $$N_{\alpha ,\beta }(d)$$ N α , β ( d</m...

In this paper, some special distance spectral properties of graphs are considered. Concretely, we recursively construct an infinite family trees with eigenvalue −1, and determine all triangle-free connected three distinct eigenvalues which the smallest one is equal to −3. Furthermore, characterize eigenvalues.

A chord of a path $P$ is an edge joining two non-adjacent vertices of $P$. A path  $P$ is called a monophonic path if it is a chordless path. A longest $x-y$ monophonic path is called an $x-y$ detour monophonic path. A  detour monophonic graphoidal cover of a graph $G$ is a collection $psi_{dm}$ of detour monophonic paths in $G$ such that every vertex of $G$ is an internal vertex  of at most on...