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Higher-order matching polynomials and d-orthogonality

2011
*
Advances in Applied Mathematics
*

The notion

doi:10.1016/j.aam.2009.12.008
fatcat:gyxafqloq5dcrefsq6q7l4fhxy
*of*d-*orthogonality*is a*generalization**of*the usual idea*of**orthogonality*for*polynomials**and*we*use*sign-reversing involutions to show that the higher-order Chebyshev (first*and*second kinds ... The*matching**polynomial**of*a*graph*is a*generating*function for coverings*of*a*graph*by disjoint edges; the higher-order*matching**polynomial*corresponds to coverings by paths. ... The number*of**matchings**of*a*graph*was*used*by Hosoya to develop his "topological index" Z , which relates chemical properties*of*hydrocarbons with*their*molecular structure. ...##
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Graph-Theoretic Solutions to Computational Geometry Problems
[article]

2009
*
arXiv
*
pre-print

Often, the efficiency

arXiv:0908.3916v1
fatcat:s4kizimp7jggliyivcqm5iokmm
*of*the algorithm depends on the special properties*of*the*graph*constructed in this way. ... Many problems in computational geometry are not stated in*graph*-theoretic terms, but can be solved efficiently by constructing an auxiliary*graph**and*performing a*graph*-theoretic algorithm on it. ... Acknowledgements This work was supported in part by NSF grant 0830403*and*by the Office*of*Naval Research under grant N00014-08-1-1015. ...##
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A digression on Hermite polynomials
[article]

2020
*
arXiv
*
pre-print

In this manuscript, we present a survey

arXiv:1901.01648v2
fatcat:237jd5r62bdsnh7q2klzaw6zdy
*of*some*general*results*of*the Hermite*polynomials**and*show a few*of**their*applications in the connection problem*of**polynomials*, probability theory*and*the combinatorics ...*Orthogonal**polynomials*are*of*fundamental importance in many fields*of*mathematics*and*science, therefore the study*of*a particular family is always relevant. ... Definition 18 A complete*match*or a perfect*match*is a j-*match*that*uses*every*vertex**of*G, the total number*of*perfect*matches*is denoted as p m (G). ...##
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Almost-Orthogonal Bases for Inner Product Polynomials
[article]

2021
*
arXiv
*
pre-print

In all three cases, we show that this expected value can be expressed in terms

arXiv:2107.00216v2
fatcat:5lboe74pgjeezgp5b7top6fb44
*of*collections*of**matchings*on the underlying*graph**of*inner products. ... We give an almost*orthogonal*basis for this vector space*of**polynomials*when the random vectors are Gaussian, spherical, or Boolean. ... We also thank Mrinalkanti Ghosh, Fernando Granha Jeronimo,*and*Madhur Tulsiani for early discussions on the*polynomials*in the context*of*Sum-*of*-Squares. ...##
###
Edge colouring models for the Tutte polynomial and related graph invariants
[article]

2007
*
arXiv
*
pre-print

) a more

arXiv:0707.2297v1
fatcat:c7m7eovelvd5lkta4t35bicxca
*general**vertex*colouring model*partition*function that includes these*polynomials**and*the principal specialization order q*of*Stanley's symmetric monochrome*polynomial*. ... For integer q>1, we*derive*edge q-colouring models for (i) the Tutte*polynomial**of*a*graph*G on the hyperbola H_q, (ii) the symmetric weight enumerator*of*the set*of*group-valued q-flows*of*G,*and*(iii ... This property allows*us*to*derive*some nonsymmetric edge q-colouring models with a*partition*function that gives the number*of*proper edge k-colourings*of*the*graph*. ...##
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Eigenvalues of graph Laplacians via rank-one perturbations
[article]

2020
*
arXiv
*
pre-print

We apply our finding to give new short proofs

arXiv:2008.01669v1
fatcat:zfjgfjfhprd7basiwznc6jgmka
*of*the spectral version*of*Kirchhoff's Matrix Tree Theorem*and*known*derivations*for the characteristic*polynomials**of*the Laplacians for several well known ... families*of**graphs*, including complete, complete multipartite,*and*threshold*graphs*. ... Acknowledgments We are grateful to Mohamed Omar for helpful*and*encouraging conversations during the early stages*of*this project. ...##
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Algorithms for diversity and clustering in social networks through dot product graphs

2015
*
Social Networks
*

u

doi:10.1016/j.socnet.2015.01.001
fatcat:tcg3nrh4uvgqrdzltym6awsrii
*and*v are connected if*and*only if*their*preferences are neither antithetical nor "*orthogonal*", that is, if*and*only if a u · a v > 0. ... The resulting*graph*is called a d-dot product*graph*. We consider diversity*and*clustering in social networks by*using*a d-dot product*graph*model for the network. ...*Using*this definition, a*polynomial*-time algorithm for Independent Set (*and*for Clique) can be easily*derived*. ...##
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Maps in locally orientable surfaces, the double coset algebra, and zonal polynomials

1996
*
Canadian Journal of Mathematics - Journal Canadien de Mathematiques
*

An expression for the genus series for (rooted) hypermaps is

doi:10.4153/cjm-1996-029-x
fatcat:nmmin45ljvhsrowtqp3pwyllxa
*derived*in terms*of*zonal*polynomials*by*using*a double coset algebra in conjunction with an encoding*of*a map as a triple*of**matchings*. ... The genus series is the*generating*series for the number*of*maps (inequivalent two-cell embeddings*of**graphs*), in locally orientable surfaces, closed*and*without boundary, with respect to*vertex*-*and*face-degrees ... This work was supported by grants from the Natural Science*and*Engineering Research Council*of*Canada. ...##
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On the Godsil -- Higman necessary condition for equitable partitions of association schemes
[article]

2013
*
arXiv
*
pre-print

Godsil

arXiv:1306.4179v1
fatcat:rvkj63uktrfehdg2gsfb5h7nha
*derived*a necessary condition for equitable*partitions**of*association schemes*and*noticed that it could be*used*to show that certain equitable*partitions*do not exist. ... Association schemes*and**their*equitable*partitions*Let V be a finite set*of*size v*and*C V ×V be the set*of*matrices over C with rows*and*columns indexed by V . ... For instance, the completely regular codes*of*the Hamming*graphs*give rise to*orthogonal*arrays,*of*the Johnson*graphs*-to combinatorial designs [4] , [8] ,*and**of*the Grassmann*graphs**of*diameter 2 ...##
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The combinatorics of associated Hermite polynomials

2009
*
European journal of combinatorics (Print)
*

We develop a combinatorial model

doi:10.1016/j.ejc.2008.05.009
fatcat:mawcythf5rg7tjxsde7bept26m
*of*the associated Hermite*polynomials**and**their*moments,*and*prove*their**orthogonality*with a sign-reversing involution. ... Several identities, linearization formulas, the moment*generating*function,*and*a second combinatorial model are also*derived*. ... The author thanks Professor Stanton for his assistance*and*patience*and*the University*of*Minnesota math department for*their*support. ...##
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Spectral folding and two-channel filter-banks on arbitrary graphs
[article]

2020
*
arXiv
*
pre-print

Therefore, many

arXiv:2010.12604v1
fatcat:w3ayynz7w5hhlmkhvma7dnqrbe
*of*the well known*orthogonal**and*bi-*orthogonal*designs can be easily adapted for*graph*signals. ... We construct GFTs satisfying a spectral folding property, which allows*us*to easily construct*orthogonal**and*bi-*orthogonal*perfect reconstruction filter-banks. ... Interestingly, our spectral domain conditions on the filters*match*those*of*[5, 6] for the normalized Laplacian*of*bipartite*graphs*,*and*therefore we can re-*use*any*of**their*filter designs, or any*of*...##
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Interlacing Properties for Hermitian Matrices Whose Graph is a Given Tree

2005
*
SIAM Journal on Matrix Analysis and Applications
*

We

doi:10.1137/s0895479804440359
fatcat:xebbohe43fel3ppa62fds2zz4i
*use*the work on*matchings**polynomials*by O.L. Heilmann*and*E.H. Lieb. ... We extend some interlacing properties*of*the eigenvalues*of*tridiagonal matrices to Hermitian matrices whose*graph*is a tree. We also give a graphical interpretation*of*the results. ... Though the*matchings**polynomial**of*a*graph*has many interesting properties, the task*of*computing this*polynomial*for a given*graph*is hard. In*general*there is no easy way*of*computing µ(G, x). ...##
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Circuit partitions and #P-complete products of inner products
[article]

2010
*
arXiv
*
pre-print

We show that q(G;k) is proportional to G's cycle

arXiv:1001.2314v1
fatcat:itvtihrr4rg3ldnqrqqm7db2my
*partition**polynomial*,*and*therefore that it is #P-complete for any k>1. ... Let G be a directed*graph*,*and*let k be a positive integer. We define q(G;k) as follows. At each*vertex*v, we place a k-dimensional complex vector x_v. ... We are grateful to PiotrŚniady for teaching*us*the sum (8),*and*to Jon Yard for introducing*us*to the Brauer algebra. ...##
###
Page 4475 of Mathematical Reviews Vol. , Issue 99g
[page]

1999
*
Mathematical Reviews
*

They also conjecture a

*general*form*of*the thresh- old*and**derive*some deterministic consequences*of**their*result with respect to the existence*of*sparse as well as critical Ramsey*graphs*. ... Summary: “The strong chromatic index*of*a*graph*G is the smallest integer k such that the edge set E(G) can be*partitioned*into k induced subgraphs*of*G which form*matchings*. ...##
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A triple lacunary generating function for Hermite polynomials
[article]

2005
*
arXiv
*
pre-print

These combinatorial interpretations are

arXiv:math/0403086v2
fatcat:5nkw2tl4fvcqfjmudof6elwh6y
*used*to prove new identities*and**generating*functions involving these*polynomials*. ... Some*of*the classical*orthogonal**polynomials*such as Hermite, Laguerre, Charlier, etc. have been shown to be the*generating**polynomials*for certain combinatorial objects. ...*derivations**of*this*generating*function can be obtained by*using*the combinatorial interpretation*of**derivatives*. ...
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