Preface 1. Introduction 2. Graph operations and modifications 3. Spectrum and structure 4. Characterizations by spectra 5. Structure and one eigenvalue 6. Spectral techniques 7. Laplacians 8. Additional topics 9. Applications Appendix Bibliography Index of symbols Index.

An Introduction to the Theory of Graph Spectra: Spectral techniques

On cohomology operations = コホモロジー作用素について

An Introduction to the Theory of Graph Spectra: Introduction

Let G be a graph and denote by Q(G)=D(G)+A(G),L(G)=D(G)-A(G) the sum and the difference between the diagonal matrix of vertex degrees and the adjacency matrix of G,respectively. In this paper,some properties of the matrix Q(G)are studied. At the same time,anecessary and sufficient condition for the equality of the spectrum of Q(G) and L(G) is given.

Some properties of the spectrum of graphs

The multiplicity of an Aα-eigenvalue: A unified approach for mixed graphs and complex unit gain graphs

On the multiplicity of α as an Aα(Γ)-eigenvalue of signed graphs with pendant vertices

The mod 2 universal Steenrod algebra Q is a homogeneous quadratic algebra closely related to the ordinary mod 2 Steenrod algebra and the Lambda algebra introduced in [1]. In this paper we show that Q is Koszul. It follows by [7] that its cohomology, being purely diagonal, is isomorphic to a completion of Q itself with respect to a suitable chain of two-sided ideals.

The Cohomology of the Universal Steenrod Algebra

Let T 4 = { ± 1 , ± i } be the subgroup of fourth roots of unity inside T , the multiplicative group of complex units. For a T 4 -gain graph Φ = ( Γ , T 4 , φ ) , we introduce gain functions on its line graph L ( Γ ) and on its subdivision graph S ( Γ ) . The corresponding gain graphs L ( Φ ) and S ( Φ ) are defined up to switching equivalence and generalize the analogous constructions for signed graphs. We discuss some spectral properties of these graphs and in particular we establish the relationship between the Laplacian characteristic polynomial of a gain graph Φ , and the adjacency characteristic polynomials of L ( Φ ) and S ( Φ ) . A suitably defined incidence matrix for T 4 -gain graphs plays an important role in this context.

Line and Subdivision Graphs Determined by T 4 -Gain Graphs

Signed bicyclic graphs minimizing the least Laplacian eigenvalue

A celebrated result by S. Priddy states the Koszulness of any locally finite homogeneous PBW-algebra, i.e. a homogeneous graded algebra having a Poincaré– Birkhoff–Witt basis. We find sufficient conditions for a non-locally finite homogeneous PBW-algebra to be Koszul, which allows us to completely determine the cohomology of the universal Steenrod algebra at any prime. Introduction. The notion of Koszul algebra, introduced by S. Priddy in [15] in particular to construct resolutions for the Steenrod algebra, has led to remarkable achievements in the study of associative algebras defined by quadratic relations. The Koszulness condition provides decisive information to solve several basic problems in that context. [14] gives a beautiful and comprehensive account of the impact of Koszul algebras in several areas of mathematics. Such algebras arise in fact in algebraic geometry, representation theory, non-commutative geometry, number theory, and obviously algebraic topology. In this paper we deal with homogeneous algebras A isomorphic to a quotient of the form T (V )/J(R), where T (V ) = ⊕ i Ti is the tensor algebra over a K-vector space V with basis X = {xi | i ∈ I}, I is a (not necessarily bounded) totally ordered set, and J(R) is the two-sided ideal of relations generated by some R ⊂ T2 = V ⊗ V . Note that all the Koszulness criteria listed for example in [9] concerning the Hilbert series of A become meaningless if I is not finite; even Priddy’s criterion, i.e. the existence of a Poincaré–Birkhoff–Witt basis [15], only holds if the algebra has an internal degree induced by a map g : ⋃ I → Z and it is locally finite with respect to length and g (see [15]). It follows that there are examples of homogeneous quadratic algebras whose Koszulness cannot be checked using directly the criteria listed in [9] and [14]: Poisson enveloping algebras of Poisson algebras with generators indexed by Z and quadratic brackets (see [11] for the definition), infinite quantum grassmannians (see 2000 Mathematics Subject Classification: 16S37, 18G15, 55S10.

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Maurizio Brunetti

Papers

On the multiplicity of α as an Aα(Γ)-eigenvalue of signed graphs with pendant vertices

The Cohomology of the Universal Steenrod Algebra

Line and Subdivision Graphs Determined by T 4 -Gain Graphs

Signed bicyclic graphs minimizing the least Laplacian eigenvalue

A Priddy-type Koszulness criterion for non-locally finite algebras