Polynomial reconstruction of signed graphs whose least eigenvalue is close to -2

2016 ◽  
Vol 31 ◽  
pp. 740-753 ◽  
Author(s):  
Slobodan Simić ◽  
Zoran Stanic
Keyword(s):  
Characteristic Polynomial ◽  
Reconstruction Problem ◽  
Signed Graphs ◽  
Simple Graphs ◽  
Least Eigenvalue ◽  
Polynomial Reconstruction ◽  
Special Classes

The polynomial reconstruction problem for simple graphs has been considered in the literature for more than forty years and is not yet resolved except for some special classes of graphs. Recently, the same problem has been put forward for signed graphs. Here, the reconstruction of the characteristic polynomial of signed graphs whose vertex-deleted subgraphs have least eigenvalue greater than $-2$ is considered.

scholarly journals Graphs Having Most of Their Eigenvalues Shared by a Vertex Deleted Subgraph

Symmetry ◽  
2021 ◽  
Vol 13 (9) ◽  
pp. 1663
Author(s):  
Alexander Farrugia
Keyword(s):  
Characteristic Polynomial ◽  
Simple Graph ◽  
Characteristic Polynomials ◽  
Reconstruction Problem ◽  
Vertex Set ◽  
Disconnected Graphs ◽  
Polynomial Reconstruction ◽  
Standing Problem

Let G be a simple graph and {1,2,…,n} be its vertex set. The polynomial reconstruction problem asks the question: given a deck P(G) containing the n characteristic polynomials of the vertex deleted subgraphs G−1, G−2, …, G−n of G, can ϕ(G,x), the characteristic polynomial of G, be reconstructed uniquely? To date, this long-standing problem has only been solved in the affirmative for some specific classes of graphs. We prove that if there exists a vertex v such that more than half of the eigenvalues of G are shared with those of G−v, then this fact is recognizable from P(G), which allows the reconstruction of ϕ(G,x). To accomplish this, we make use of determinants of certain walk matrices of G. Our main result is used, in particular, to prove that the reconstruction of the characteristic polynomial from P(G) is possible for a large subclass of disconnected graphs, strengthening a result by Sciriha and Formosa.

scholarly journals Joining Forces for Reconstruction Inverse Problems

Symmetry ◽  
2021 ◽  
Vol 13 (9) ◽  
pp. 1687
Author(s):  
Irene Sciriha
Keyword(s):  
Inverse Problem ◽  
Inverse Problems ◽  
Graph Matching ◽  
Isomorphism Problem ◽  
Reconstruction Problem ◽  
Graph Invariants ◽  
Minimal Set ◽  
Disconnected Graphs ◽  
Polynomial Reconstruction ◽  
Spectral Inverse Problem

A spectral inverse problem concerns the reconstruction of parameters of a parent graph from prescribed spectral data of subgraphs. Also referred to as the P–NP Isomorphism Problem, Reconstruction or Exact Graph Matching, the aim is to seek sets of parameters to determine a graph uniquely. Other related inverse problems, including the Polynomial Reconstruction Problem (PRP), involve the recovery of graph invariants. The PRP seeks to extract the spectrum of a graph from the deck of cards each showing the spectrum of a vertex-deleted subgraph. We show how various algebraic methods join forces to reconstruct a graph or its invariants from a minimal set of restricted eigenvalue-eigenvector information of the parent graph or its subgraphs. We show how functions of the entries of eigenvectors of the adjacency matrix A of a graph can be retrieved from the spectrum of eigenvalues of A. We establish that there are two subclasses of disconnected graphs with each card of the deck showing a common eigenvalue. These could occur as possible counter examples to the positive solution of the PRP.

scholarly journals Spectrum and L-spectrum of the power graph and its main supergraph for certain finite groups

Filomat ◽  
2017 ◽  
Vol 31 (16) ◽  
pp. 5323-5334 ◽  
Author(s):  
Asma Hamzeh ◽  
Ali Ashrafi
Keyword(s):  
Finite Group ◽  
Characteristic Polynomial ◽  
Finite Groups ◽  
The Other ◽  
Laplacian Spectrum ◽  
Simple Graphs ◽  
Power Graph ◽  
Vertex Set ◽  
A Finite Group

Let G be a finite group. The power graph P(G) and its main supergraph S(G) are two simple graphs with the same vertex set G. Two elements x,y ? G are adjacent in the power graph if and only if one is a power of the other. They are joined in S(G) if and only if o(x)|o(y) or o(y)|o(x). The aim of this paper is to compute the characteristic polynomial of these graph for certain finite groups. As a consequence, the spectrum and Laplacian spectrum of these graphs for dihedral, semi-dihedral, cyclic and dicyclic groups were computed.

scholarly journals Signed graphs with least eigenvalue <−2

1992 ◽  
Vol 13 (3) ◽  
pp. 219-220 ◽  
Author(s):  
N.M. Singhi ◽  
G.R. Vijayakumar
Keyword(s):  
Signed Graphs ◽  
Least Eigenvalue

scholarly journals Polynomial reconstruction of signed graphs

2016 ◽  
Vol 501 ◽  
pp. 390-408 ◽  
Author(s):  
Slobodan K. Simić ◽  
Zoran Stanić
Keyword(s):  
Signed Graphs ◽  
Polynomial Reconstruction
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