# 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

- Mathematics
- 2013

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

- Mathematics, Computer ScienceElectron. J. Comb.
- 2017

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

- Mathematics
- 2005

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

- Mathematics
- 2013

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

- Mathematics
- 2017

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

- Mathematics
- 1994

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

- MathematicsLinear Algebra and its Applications
- 2019

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

- Computer Science, MathematicsJ. Comb. Optim.
- 2020

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

- Mathematics, Computer ScienceSIAM J. Matrix Anal. Appl.
- 2008

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☆

- Mathematics
- 2007

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…