Strongly Regular Graphs

Strongly regular graphs lie at the intersection of statistical design, group theory, finite geometry, information and coding theory, and extremal combinatorics. This monograph collects all the major known results together for the first time in book form, creating an invaluable text that researchers in algebraic combinatorics and related areas will refer to for years to come. The book covers the theory of strongly regular graphs, polar graphs, rank 3 graphs associated to buildings and Fischer groups, cyclotomic graphs, two-weight codes and graphs related to combinatorial configurations such as Latin squares, quasi-symmetric designs and spherical designs. It gives the complete classification of rank 3 graphs, including some new constructions. More than 100 graphs are treated individually. Some unified and streamlined proofs are featured, along with original material including a new approach to the (affine) half spin graphs of rank 5 hyperbolic polar spaces.

Two-weight and three-weight linear codes constructed from Weil sums

<p style='text-indent:20px;'>Linear codes with few weights are widely used in strongly regular graphs, secret sharing schemes, association schemes and authentication codes. In this paper, we construct several two-weight and three-weight linear codes over finite fields by choosing suitable different defining sets. We also give some examples and some of the codes are optimal or almost optimal. Their applications to secret sharing schemes are also investigated.</p>

Fast Algorithms for General Spin Systems on Bipartite Expanders

A spin system is a framework in which the vertices of a graph are assigned spins from a finite set. The interactions between neighbouring spins give rise to weights, so a spin assignment can also be viewed as a weighted graph homomorphism. The problem of approximating the partition function (the aggregate weight of spin assignments) or of sampling from the resulting probability distribution is typically intractable for general graphs. In this work, we consider arbitrary spin systems on bipartite expander Δ-regular graphs, including the canonical class of bipartite random Δ-regular graphs. We develop fast approximate sampling and counting algorithms for general spin systems whenever the degree and the spectral gap of the graph are sufficiently large. Roughly, this guarantees that the spin system is in the so-called low-temperature regime. Our approach generalises the techniques of Jenssen et al. and Chen et al. by showing that typical configurations on bipartite expanders correspond to “bicliques” of the spin system; then, using suitable polymer models, we show how to sample such configurations and approximate the partition function in Õ( n 2 ) time, where n is the size of the graph.

Cartesian Products of Some Regular Graphs Admitting Antimagic Labeling for Arbitrary Sets of Real Numbers

An edge labeling of graph G with labels in A is an injection from E G to A , where E G is the edge set of G , and A is a subset of ℝ . A graph G is called ℝ -antimagic if for each subset A of ℝ with A = E G , there is an edge labeling with labels in A such that the sums of the labels assigned to edges incident to distinct vertices are different. The main result of this paper is that the Cartesian products of complete graphs (except K 1 ) and cycles are ℝ -antimagic.

SHILLA GRAPHS WITH \(b=5\) AND \(b=6\)

A \(Q\)-polynomial Shilla graph with \(b = 5\) has intersection arrays \(\{105t,4(21t+1),16(t+1); 1,4 (t+1),84t\}\), \(t\in\{3,4,19\}\). The paper proves that distance-regular graphs with these intersection arrays do not exist. Moreover, feasible intersection arrays of \(Q\)-polynomial Shilla graphs with \(b = 6\) are found.

Distance-Regular Graphs with Intersection Arrays {7,6,6;1,1,2} and {42,30,2;1,10,36} Do not Exist

Пусть $\Gamma$ - дистанционно регулярный граф диаметра 3 без треугольников, $u$ - вершина графа $\Gamma$, $\Delta^i=\Gamma_i(u)$ и $\Sigma^i=\Delta^i_{2,3}$. Тогда $\Sigma^i$ - регулярный граф без 3-коклик степени $k'=k_i-a_i-1$ на $v'=k_i$ вершинах. Заметим, что для несмежных вершин $y,z\in \Sigma^i$ имеем $\Sigma^i=\{y,z\}\cup \Sigma^i(y)\cup \Sigma^i(z)$. Поэтому для $\mu'=|\Sigma^i(y)\cap \Sigma^i(z)|$ имеем равенство $v'=2k'+2-\mu'$. Отсюда граф $\Sigma$ является кореберно регулярным с параметрами $(v',k',\mu')$. В работе доказано, что дистанционно регулярный граф с массивом пересечений $\{7,6,6;1,1,2\}$ не существует. В статье М. С. Нировой "On distance-regular graphs with $\theta_2=-1$" показано, что если существует сильно регулярный граф с параметрами $(176,49,12,14)$, в котором окрестности вершин являются $7\times 7$-решетками, то существует и дистанционно регулярный граф с массивом пересечений $\{7,6,6;1,1,2\}$. М.~П. Голубятников заметил, что для дистанционно регулярного графа $\Gamma$ с массивом пересечений $\{7,6,6;1,1,2\}$ граф $\Gamma_2$ является дистанционно регулярным с массивом пересечений $\{42,30,2;1,10,36\}$. С помощью этого результата и вычисления тройных чисел пересечений доказано, что дистанционно регулярные графы с массивами пересечений $\{7,6,6;1,1,2\}$ и $\{42,30,2;1,10,36\}$ не существуют.