Equiangular lines in the real space $${\mathbb {R}}^d$$ R d
- PDF / 221,293 Bytes
- 9 Pages / 439.37 x 666.142 pts Page_size
- 83 Downloads / 175 Views
(0123456789().,-volV) (0123456789().,-volV)
SURVEY PAPER
Equiangular lines in the real space Rd Sharad Sane1 Received: 19 November 2019 / Accepted: 18 January 2020 Ó Forum D’Analystes, Chennai 2020
Abstract This short note describes yet another interplay between linear algebra and graph theory, in connection with a problem of equiangular lines. Keywords Equiangular lines Absolute bound Relative bound Mathematics Subject Classification 05C50 05C90
1 Introduction and preliminaries on graphs I thank the organisers of the SHKFEST for asking me to submit a note on a suitable mathematical topic for the workshop and the conference in honour of Professor Sudhir H. Kulkarni who turned 65 in 2018. I have known SHK (as he is popularly called by his colleagues and admirers including me) for a long time and that was the main reason of choosing to write on this topic of equiangular lines in the real space, though my own area of research happens to be combinatorics. In a surprising manner this problem finally turns out to be a problem of graph theory and more importantly that of linear algebra. The latter, I always thought, has been a formidable meeting ground between combinatorics and finite situations on the one hand and analysis on the other. I presume that this will form a decent tribute to SHK who successfully worked in analysis related areas for several decades. We now give a quick review of some preliminaries on simple graphs and some interesting simple graphs in particular, that will be used in the next section. These preliminaries can be found on any of the standard quoted graph theory texts in the references [2, page 6; also 3, page 452].
& Sharad Sane [email protected]; [email protected] 1
Chennai Mathematical Institute, Chennai 603103, India
123
S. Sane
The exposition exclusively deals with a simple graph C with n vertices and with m edges. The adjacency matrix A of C and the (0,1)-incidence matrix M (of order n m) of C are connected by the following matrix equation. MM t ¼ diagðd1 ; d2 ; . . .; dn Þ þ A; where di s are the vertex degrees. Let L ¼ LðCÞ denote the line graph of C. Here LðCÞ has edges of C as its vertices and with two edges adjacent if they share a common vertex. Then: M t M ¼ 2I þ AðLÞ; where A(L) denotes the adjacency matrix of L. Since an adjacency matrix is determined by a graph uniquely up to permutation of vertices, it makes sense (with a little abuse of language) to talk of eigenvalues of a graph rather than eigenvalues of its adjacency matrix. In the standard graph theory framework, the term spectrum of a graph C refers to its eigenvalues along with multiplicities (written in a decreasing order of the eigenvalues). Here the first row gives the eigenvalues and the second row corresponding multiplicities. The spectrum of the complete graph Kn is given by: n1 1 specðKn Þ ¼ : 1 n1 As has already been explained, this simply means that n 1 is an eigenvalue with multiplicity 1 and 1 the other eigenvalue with multiplicity n 1. Let C be k-regular with 1 k n 1. Then k i
Data Loading...
