The maximum multiplicity and the two largest multiplicities of eigenvalues in a Hermitian matrix whose graph is a tree

Connections between vital linkages and zero forcing are established. Specifically, the notion of a rigid linkage is introduced as a special kind of unique linkage and it is shown that spanning forcing paths of a zero forcing process form a spanning rigid linkage and thus a vital linkage. A related generalization of zero forcing that produces a rigid linkage via a coloring process is developed. One of the motivations for introducing zero forcing is to provide an upper bound on the maximum multiplicity of an eigenvalue among the real symmetric matrices described by a graph. Rigid linkages and a related notion of rigid shortest linkages are utilized to obtain bounds on the multiplicities of eigenvalues of this family of matrices.

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The change in multiplicity of an eigenvalue of a Hermitian matrix associated with the removal of an edge from its graph

Discrete Mathematics ◽

10.1016/j.disc.2010.10.010 ◽

2011 ◽

Vol 311 (2-3) ◽

pp. 166-170 ◽

Cited By ~ 5

Author(s):

Charles R. Johnson ◽

Paul R. McMichael

Keyword(s):

Hermitian Matrix ◽

Multiplicity Of An Eigenvalue

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Genus expansion of matrix models and ћ expansion of KP hierarchy

Journal of High Energy Physics ◽

10.1007/jhep12(2020)038 ◽

2020 ◽

Vol 2020 (12) ◽

Author(s):

A. Andreev ◽

A. Popolitov ◽

A. Sleptsov ◽

A. Zhabin

Keyword(s):

Matrix Models ◽

Matrix Model ◽

Special Interest ◽

Hermitian Matrix ◽

Geometric Meaning ◽

Kp Hierarchy ◽

Hurwitz Numbers ◽

Kp Equations ◽

Genus Expansion ◽

Hermitian Matrix Model

Abstract We study ћ expansion of the KP hierarchy following Takasaki-Takebe [1] considering several examples of matrix model τ-functions with natural genus expansion. Among the examples there are solutions of KP equations of special interest, such as generating function for simple Hurwitz numbers, Hermitian matrix model, Kontsevich model and Brezin-Gross-Witten model. We show that all these models with parameter ћ are τ-functions of the ћ-KP hierarchy and the expansion in ћ for the ћ-KP coincides with the genus expansion for these models. Furthermore, we show a connection of recent papers considering the ћ-formulation of the KP hierarchy [2, 3] with original Takasaki-Takebe approach. We find that in this approach the recovery of enumerative geometric meaning of τ-functions is straightforward and algorithmic.

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The location of classified edges due to the change in the geometric multiplicity of an eigenvalue in a tree

Special Matrices ◽

10.1515/spma-2019-0019 ◽

2019 ◽

Vol 7 (1) ◽

pp. 257-262

Author(s):

Kenji Toyonaga

Keyword(s):

Symmetric Matrix ◽

Symmetric Matrices ◽

Sufficient Condition ◽

Necessary And Sufficient Condition ◽

Geometric Multiplicity ◽

Necessary And Sufficient ◽

Multiplicity Of An Eigenvalue

Abstract Given a combinatorially symmetric matrix A whose graph is a tree T and its eigenvalues, edges in T can be classified in four categories, based upon the change in geometric multiplicity of a particular eigenvalue, when the edge is removed. We investigate a necessary and sufficient condition for each classification of edges. We have similar results as the case for real symmetric matrices whose graph is a tree. We show that a g-2-Parter edge, a g-Parter edge and a g-downer edge are located separately from each other in a tree, and there is a g-neutral edge between them. Furthermore, we show that the distance between a g-downer edge and a g-2-Parter edge or a g-Parter edge is at least 2 in a tree. Lastly we give a combinatorially symmetric matrix whose graph contains all types of edges.

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The planar limit of $$ \mathcal{N} $$ = 2 chiral correlators

Journal of High Energy Physics ◽

10.1007/jhep08(2021)032 ◽

2021 ◽

Vol 2021 (8) ◽

Author(s):

Bartomeu Fiol ◽

Alan Rios Fukelman

Keyword(s):

Free Energy ◽

Infinite Number ◽

Correlation Functions ◽

Hermitian Matrix ◽

Hooft Coupling ◽

Planar Limit ◽

Single Trace ◽

Hermitian Matrix Model ◽

Combinatorial Expression ◽

Do So

Abstract We derive the planar limit of 2- and 3-point functions of single-trace chiral primary operators of $$ \mathcal{N} $$ N = 2 SQCD on S4, to all orders in the ’t Hooft coupling. In order to do so, we first obtain a combinatorial expression for the planar free energy of a hermitian matrix model with an infinite number of arbitrary single and double trace terms in the potential; this solution might have applications in many other contexts. We then use these results to evaluate the analogous planar correlation functions on ℝ4. Specifically, we compute all the terms with a single value of the ζ function for a few planar 2- and 3-point functions, and conjecture general formulas for these terms for all 2- and 3-point functions on ℝ4.

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A Schur-Cohn theorem for matrix polynomials

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091500004806 ◽

1990 ◽

Vol 33 (3) ◽

pp. 337-366 ◽

Cited By ~ 6

Author(s):

Harry Dym ◽

Nicholas Young

Keyword(s):

Unit Circle ◽

Matrix Polynomial ◽

Hermitian Matrix ◽

Krein Space ◽

Gram Matrix ◽

Natural Basis ◽

Test Matrix ◽

Matrix Polynomials ◽

Rational Vector ◽

Square Matrix

Let N(λ) be a square matrix polynomial, and suppose det N is a polynomial of degree d. Subject to a certain non-singularity condition we construct a d by d Hermitian matrix whose signature determines the numbers of zeros of N inside and outside the unit circle. The result generalises a well known theorem of Schur and Cohn for scalar polynomials. The Hermitian “test matrix” is obtained as the inverse of the Gram matrix of a natural basis in a certain Krein space of rational vector functions associated with N. More complete results in a somewhat different formulation have been obtained by Lerer and Tismenetsky by other methods.

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