The NIEP

The nonnegative inverse eigenvalue problem (NIEP) asks which lists of $n$ complex numbers (counting multiplicity) occur as the eigenvalues of some $n$-by-$n$ entry-wise nonnegative matrix. The NIEP has a long history and is a known hard (perhaps the hardest in matrix analysis?) and sought after problem. Thus, there are many subproblems and relevant results in a variety of directions. We survey most work on the problem and its several variants, with an emphasis on recent results, and include 130 references. The survey is divided into: a) the single eigenvalue problems; b) necessary conditions; c) low dimensional results; d) sufficient conditions; e) appending 0's to achieve realizability; f) the graph NIEP's; g) Perron similarities; and h) the relevance of Jordan structure

LpL^p solvability of the Stationary Stokes problem on domains with conical singularity in any dimension

The Dirichlet boundary value problem for the Stokes operator with $L^p$ data in any dimension on domains with conical singularity (not necessary a Lipschitz graph) is considered. We establish the solvability of the problem for all $p\in (2-\varepsilon,\infty]$ and also its solvability in $C(\overline{D})$ for the data in $C(\partial D)

Isospectral deformations of the Dirac operator

We give more details about an integrable system in which the Dirac operator D=d+d^* on a finite simple graph G or Riemannian manifold M is deformed using a Hamiltonian system D'=[B,h(D)] with B=d-d^* + i b. The deformed operator D(t) = d(t) + b(t) + d(t)^* defines a new exterior derivative d(t) and a new Dirac operator C(t) = d(t) + d(t)^* and Laplacian M(t) = d(t) d(t)^* + d(t)* d(t) and so a new distance on G or a new metric on M.Comment: 32 pages, 8 figure

On the Existence of Non-golden Signed Graphs

A signed graph is a pair Γ = (G,σ), where G = (V(G),E(G)) is a graph and σ : E(G)→{+1,−1} is the sign function on the edges of G. For a signed graph we consider the least eigenvector λ(Γ) of the Laplacian matrix defined as L(Γ) = D(G)−A(Γ), where D(G) is the matrix of vertices degrees of G and A(Γ) is the signed adjacency matrix. An unbalanced signed bicyclic graph is said to be golden if it is switching equivalent to a graph Γ satisfying the following property: there exists a cycle C in Γ and a λ(Γ)-eigenvector x such that the unique negative edge pq of Γ belongs to C and detects the minimum of the set Sx(Γ,C) = { |xrxs| | rs ∈ E(C) }. In this paper we show that non-golden bicyclic graphs with frustration index 1 exist for each n ≥ 5

The domination number and the least QQ-eigenvalue

A vertex set $D$ of a graph $G$ is said to be a dominating set if every vertex of $V(G)\setminus D$ is adjacent to at least a vertex in $D$, and the domination number $\gamma(G)$ ($\gamma$, for short) is the minimum cardinality of all dominating sets of $G$. For a graph, the least $Q$-eigenvalue is the least eigenvalue of its signless Laplacian matrix. In this paper, for a nonbipartite graph with both order $n$ and domination number $\gamma$, we show that $n\geq 3\gamma-1$, and show that it contains a unicyclic spanning subgraph with the same domination number $\gamma$. By investigating the relation between the domination number and the least $Q$-eigenvalue of a graph, we minimize the least $Q$-eigenvalue among all the nonbipartite graphs with given domination number.Comment: 13 pages, 3 figure

A family of diameter-based eigenvalue bounds for quantum graphs

We establish a sharp lower bound on the first non-trivial eigenvalue of the Laplacian on a metric graph equipped with natural (i.e., continuity and Kirchhoff) vertex conditions in terms of the diameter and the total length of the graph. This extends a result of, and resolves an open problem from, [J. B. Kennedy, P. Kurasov, G. Malenov\'a and D. Mugnolo, Ann. Henri Poincar\'e 17 (2016), 2439--2473, Section 7.2], and also complements an analogous lower bound for the corresponding eigenvalue of the combinatorial Laplacian on a discrete graph. We also give a family of corresponding lower bounds for the higher eigenvalues under the assumption that the total length of the graph is sufficiently large compared with its diameter. These inequalities are sharp in the case of trees.Comment: Substantial revision of v1. The main result, originally for the first eigenvalue, has been generalised to the higher ones. The title has been changed and the proofs substantially reorganised to reflect the new result, and a section containing concluding remarks has been adde