scholarly journals Ordered multiplicity inverse eigenvalue problem for graphs on six vertices

2021 ◽  
Vol 37 ◽  
pp. 316-358
Author(s):  
John Ahn ◽  
Christine Alar ◽  
Beth Bjorkman ◽  
Steve Butler ◽  
Joshua Carlson ◽  
...  
Keyword(s):  
Eigenvalue Problem ◽  
Complete Solution ◽  
Inverse Eigenvalue Problem ◽  
Symmetric Matrices ◽  
Significant Progress ◽  
Connected Graphs ◽  
Inverse Eigenvalue ◽  
Eigenvalue Multiplicities ◽  
Do So

For a graph $G$, we associate a family of real symmetric matrices, $\mathcal{S}(G)$, where for any $M \in \mathcal{S}(G)$, the location of the nonzero off-diagonal entries of $M$ is governed by the adjacency structure of $G$. The ordered multiplicity Inverse Eigenvalue Problem of a Graph (IEPG) is concerned with finding all attainable ordered lists of eigenvalue multiplicities for matrices in $\mathcal{S}(G)$. For connected graphs of order six, we offer significant progress on the IEPG, as well as a complete solution to the ordered multiplicity IEPG. We also show that while $K_{m,n}$ with $\min(m,n)\ge 3$ attains a particular ordered multiplicity list, it cannot do so with arbitrary spectrum.

The combinatorial inverse eigenvalue problem II: all cases for small graphs

2014 ◽  
Vol 27 ◽  
Author(s):  
Wayne Barrett ◽  
Curtis Nelson ◽  
John Sinkovic ◽  
Tianyi Yang
Keyword(s):  
Eigenvalue Problem ◽  
Undirected Graph ◽  
Complete Solution ◽  
Inverse Eigenvalue Problem ◽  
Pendant Vertex ◽  
Real Numbers ◽  
Connected Graphs ◽  
The Matrix ◽  
Inverse Eigenvalue

Let G be a simple undirected graph on n vertices and let S(G) be the class of real symmetric n by n matrices whose nonzero off-diagonal entries correspond to the edges of G. Given 2n - 1 real numbers \lambda_1\geq \mu_1 \geq \lambda_2 \geq \mu_2 \geq \cdots \geq \lambda_{n-1} \geq \mu_{n-1} \geq \lambda_{n-1}, and a vertex v of G, the question is addressed of whether or not there exists A in S(G) with eigenvalues \lambda_1, \ldots, \lambda_ n such that A(v) has eigenvalues \mu_1, \ldots, \mu_{n-1}, where A(v) denotes the matrix with vth row and column deleted. A complete solution can be given for the path on n vertices with v a pendant vertex and also for the star on n vertices with v the dominating vertex. The main result is a complete solution to this "\lambda, \mu" problem for all connected graphs on 4 vertices.

The Inverse Eigenvalue Problem for Symmetric Doubly Arrow Matrices

2013 ◽  
Vol 444-445 ◽  
pp. 625-627
Author(s):  
Kan Ming Wang ◽  
Zhi Bing Liu ◽  
Xu Yun Fei
Keyword(s):  
Eigenvalue Problem ◽  
Special Kind ◽  
Inverse Eigenvalue Problem ◽  
Symmetric Matrices ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
The Real ◽  
Inverse Eigenvalue ◽  
Necessary And Sufficient

In this paper we present a special kind of real symmetric matrices: the real symmetric doubly arrow matrices. That is, matrices which look like two arrow matrices, forward and backward, with heads against each other at the station, . We study a kind of inverse eigenvalue problem and give a necessary and sufficient condition for the existence of such matrices.

scholarly journals Achievable multiplicity partitions in the inverse eigenvalue problem of a graph

2019 ◽  
Vol 7 (1) ◽  
pp. 276-290
Author(s):  
Mohammad Adm ◽  
Shaun Fallat ◽  
Karen Meagher ◽  
Shahla Nasserasr ◽  
Sarah Plosker ◽  
...  
Keyword(s):  
Eigenvalue Problem ◽  
Inverse Eigenvalue Problem ◽  
Symmetric Matrices ◽  
Multipartite Graphs ◽  
Complete Multipartite Graphs ◽  
Inverse Eigenvalue ◽  
Complete Multipartite

Abstract Associated to a graph G is a set 𝒮(G) of all real-valued symmetric matrices whose off-diagonal entries are nonzero precisely when the corresponding vertices of the graph are adjacent, and the diagonal entries are free to be chosen. If G has n vertices, then the multiplicities of the eigenvalues of any matrix in 𝒮 (G) partition n; this is called a multiplicity partition. We study graphs for which a multiplicity partition with only two integers is possible. The graphs G for which there is a matrix in 𝒮 (G) with partitions [n − 2, 2] have been characterized. We find families of graphs G for which there is a matrix in 𝒮 (G) with multiplicity partition [n − k, k] for k ≥ 2. We focus on generalizations of the complete multipartite graphs. We provide some methods to construct families of graphs with given multiplicity partitions starting from smaller such graphs. We also give constructions for graphs with matrix in 𝒮 (G) with multiplicity partition [n − k, k] to show the complexities of characterizing these graphs.

scholarly journals Explicit Solution of the Inverse Eigenvalue Problem of Real Symmetric Matrices and Its Application to Electrical Network Synthesis

2008 ◽  
Vol 2008 ◽  
pp. 1-25 ◽  
Author(s):  
D. B. Kandić ◽  
B. D. Reljin
Keyword(s):  
Eigenvalue Problem ◽  
Common Ground ◽  
Explicit Construction ◽  
Inverse Eigenvalue Problem ◽  
Electrical Network ◽  
Symmetric Matrices ◽  
Network Synthesis ◽  
Sign Patterns ◽  
Specific Sign ◽  
Inverse Eigenvalue

A novel procedure for explicit construction of the entries of real symmetric matrices with assigned spectrum and the entries of the corresponding orthogonal modal matrices is presented. The inverse eigenvalue problem of symmetric matrices with some specific sign patterns (including hyperdominant one) is explicitly solved too. It has been shown to arise thereof a possibility of straightforward solving the inverse eigenvalue problem of symmetric hyperdominant matrices with assigned nonnegative spectrum. The results obtained are applied thereafter in synthesis of driving-point immittance functions of transformerless, common-ground, two-element-kindRLCnetworks and in generation of their equivalent realizations.

scholarly journals The Diagonalizable Nonnegative Inverse Eigenvalue Problem

2018 ◽  
Vol 6 (1) ◽  
pp. 273-281 ◽  
Author(s):  
Anthony G Cronin ◽  
Thomas J. Laffey
Keyword(s):  
Normal Form ◽  
Eigenvalue Problem ◽  
Inverse Eigenvalue Problem ◽  
Symmetric Matrices ◽  
Nonnegative Matrices ◽  
Jordan Normal Form ◽  
Inverse Eigenvalue ◽  
Diagonalizable Matrix

Abstract In this articlewe provide some lists of real numberswhich can be realized as the spectra of nonnegative diagonalizable matrices but which are not the spectra of nonnegative symmetric matrices. In particular, we examine the classical list σ = (3 + t, 3 − t, −2, −2, −2) with t ≥ 0, and show that 0 is realizable by a nonnegative diagonalizable matrix only for t ≥ 1. We also provide examples of lists which are realizable as the spectra of nonnegative matrices, but not as the spectra of nonnegative diagonalizable matrices by examining the Jordan Normal Form

scholarly journals Solvability of Inverse Eigenvalue Problem for Dense Singular Symmetric Matrices

2013 ◽  
Vol 03 (01) ◽  
pp. 14-19
Author(s):  
Anthony Y. Aidoo ◽  
Kwasi Baah Gyamfi ◽  
Joseph Ackora-Prah ◽  
Francis T. Oduro
Keyword(s):  
Eigenvalue Problem ◽  
Inverse Eigenvalue Problem ◽  
Symmetric Matrices ◽  
Inverse Eigenvalue
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