• Corpus ID: 239024379

On the minimum number of distinct eigenvalues of a threshold graph

@inproceedings{Fallat2021OnTM,
  title={On the minimum number of distinct eigenvalues of a threshold graph},
  author={Shaun M. Fallat and Seyed Ahmad Mojallal},
  year={2021}
}
For a graph G, we associate a family of real symmetric matrices, S (G), where for any A ∈ S (G), the location of the nonzero off-diagonal entries of A are governed by the adjacency structure of G. Let q(G) be the minimum number of distinct eigenvalues over all matrices in S (G). In this work, we give a characterization of all connected threshold graphs G with q(G) = 2. Moreover, we study the values of q(G) for connected threshold graphs with trace 2, 3, n − 2, n − 3, where n is the order of… 

References

SHOWING 1-10 OF 18 REFERENCES
Minimum number of distinct eigenvalues of graphs
The minimum number of distinct eigenvalues, taken over all real symmetric matrices compatible with a given graph G, is denoted by q(G). Using other parameters related to G, bounds for q(G) are proven
Generalizations of the Strong Arnold Property and the Minimum Number of Distinct Eigenvalues of a Graph
TLDR
Two extensions are devised that target a better understanding of all possible spectra and their associated multiplicities, and are referred to as the Strong Spectral Property and the Strong Multiplicity Property.
SPECTRAL GRAPH THEORY AND THE INVERSE EIGENVALUE PROBLEM OF A GRAPH
Spectral Graph Theory is the study of the spectra of certain matrices defined from a given graph, including the adjacency matrix, the Laplacian matrix and other related matrices. Graph spectra have
On the adjacency matrix of a threshold graph
Abstract A threshold graph on n vertices is coded by a binary string of length n − 1 . We obtain a formula for the inertia of (the adjacency matrix of) a threshold graph in terms of the code of the
Ordered multiplicity inverse eigenvalue problem for graphs on six vertices
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 $M$ are governed by the
Degree maximal graphs are Laplacian integral
Abstract An integer sequence ( d ) = ( d 1 , d 2 ,…, d n ) is graphic if there is a graph whose degree sequence is ( d ). A graph G is maximal if its degree sequence is majorized by no other graphic
A Nordhaus–Gaddum conjecture for the minimum number of distinct eigenvalues of a graph
We propose a Nordhaus-Gaddum conjecture for $q(G)$, the minimum number of distinct eigenvalues of a symmetric matrix corresponding to a graph $G$: for every graph $G$ excluding four exceptions, we
On the zero forcing number of a graph involving some classical parameters
TLDR
All the connected graphs G are identified and the equality holds for threshold graphs with Z (G) = p (G)-ex(G) + 2 Φ ( G ) - 2 and the inequality is shown.
On the Minimum Rank Among Positive Semidefinite Matrices with a Given Graph
TLDR
Upper and lower bounds for the minimum rank of all matrices in the set of all positive semidefinite matrices whose graph is G are given and used to determineoperatorname(G) for some well-known graphs.
The minimum rank of symmetric matrices described by a graph: A survey☆
Abstract The minimum rank of a simple graph G is defined to be the smallest possible rank over all symmetric real matrices whose ijth entry (for i ≠ j ) is nonzero whenever { i , j } is an edge in G
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