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### Higher-order matching polynomials and d-orthogonality

Dan Drake
The notion 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.  ...

### Graph-Theoretic Solutions to Computational Geometry Problems [article]

David Eppstein
2009 arXiv   pre-print
Often, the efficiency 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.  ...

### A digression on Hermite polynomials [article]

Keith Y. Patarroyo
2020 arXiv   pre-print
In this manuscript, we present a survey 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).  ...

### Almost-Orthogonal Bases for Inner Product Polynomials [article]

Chris Jones, Aaron Potechin
2021 arXiv   pre-print
In all three cases, we show that this expected value can be expressed in terms 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]

Andrew J. Goodall
2007 arXiv   pre-print
) a more 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.  ...

### Eigenvalues of graph Laplacians via rank-one perturbations [article]

Steven Klee, Matthew T. Stamps
2020 arXiv   pre-print
We apply our finding to give new short proofs 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.  ...

### Algorithms for diversity and clustering in social networks through dot product graphs

Matthew Johnson, Daniël Paulusma, Erik Jan van Leeuwen
2015 Social Networks
u 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.  ...

### Maps in locally orientable surfaces, the double coset algebra, and zonal polynomials

I. P. Goulden, D. M. Jackson
An expression for the genus series for (rooted) hypermaps is 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.  ...

### On the Godsil -- Higman necessary condition for equitable partitions of association schemes [article]

Alexander L. Gavrilyuk, Ivan Yu. Mogilnykh
2013 arXiv   pre-print
Godsil 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  ,  , and of the Grassmann graphs of diameter 2  ...

### The combinatorics of associated Hermite polynomials

Dan Drake
2009 European journal of combinatorics (Print)
We develop a combinatorial model 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.  ...

### Spectral folding and two-channel filter-banks on arbitrary graphs [article]

Eduardo Pavez, Benjamin Girault, Antonio Ortega, Philip A. Chou
2020 arXiv   pre-print
Therefore, many 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  ...

### Interlacing Properties for Hermitian Matrices Whose Graph is a Given Tree

C. M. da Fonseca
2005 SIAM Journal on Matrix Analysis and Applications
We 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).  ...

### Circuit partitions and #P-complete products of inner products [article]

Cristopher Moore, Alexander Russell
2010 arXiv   pre-print
We show that q(G;k) is proportional to G's cycle 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.  ...

### A triple lacunary generating function for Hermite polynomials [article]

Ira M. Gessel, Pallavi Jayawant
2005 arXiv   pre-print
These combinatorial interpretations are 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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