A graph is said to be exceptional if it is connected, has least eigenvalue greater than or equal to −2, and is not a generalized line graph. Such graphs are known to be representable in the exceptional root system E 8. The 473 maximal exceptional graphs have been found by computer, and the 467 with maximal degree 28 have been characterized. Here we construct the remaining six maximal exceptiona...

let $g=(v‎, ‎e)$ be a graph with $p$ vertices and $q$ edges‎. ‎an emph{acyclic‎ ‎graphoidal cover} of $g$ is a collection $psi$ of paths in $g$‎ ‎which are internally-disjoint and cover each edge of the graph‎ ‎exactly once‎. ‎let $f‎: ‎vrightarrow {1‎, ‎2‎, ‎ldots‎, ‎p}$ be a bijective‎ ‎labeling of the vertices of $g$‎. ‎let $uparrow!g_f$ be the‎ ‎directed graph obtained by orienting the...

In this note a new measure of irregularity of a simple undirected graph G is introduced. It is named the total irregularity of a graph and is defined as irrt(G) = 1 2 ∑ u,v∈V (G) |dG(u)− dG(v)| , where dG(u) denotes the degree of a vertex u ∈ V (G). The graphs with maximal total irregularity are determined. It is also shown that among all trees of same order the star graph has the maximal total...

A graph G is well-covered if all its maximal independent sets are of the same cardinality. Assume that a weight function w is defined on its vertices. Then G is w-well-covered if all maximal independent sets are of the same weight. For every graph G, the set of weight functions w such that G is w-well-covered is a vector space. Given an input claw-free graph G, we present an O (

The recognition problem for visibility graphs of simple polygons is not known to be in NP, nor is it known to be NP-hard. It is, however, known to be in PSPACE. Further, every such visibility graph can be dismantled as a sequence of visibility graphs of convex fans. Any nondegenerate configuration of n points can be associated with a maximal chain in the weak Bruhat order of the symmetric group...

Let k be a positive integer, a subset Q of the set of vertices of a graph G is k-dependent in G if each vertex of Q has no more than k neighbours in Q. We present a parallel algorithm which computes a maximal k-dependent set in a graph on n nodes in time O(log 4 n) on an EREW PRAM with O(n 2) processors. In this way, we establish the membership of the problem of constructing a maximal k-depende...

The algorithm MLS (Maximal Label Search) is a graph search algorithm that generalizes the algorithms Maximum Cardinality Search (MCS), Lexicographic Breadth-First Search (LexBFS), Lexicographic Depth-First Search (LexDFS) and Maximal Neighborhood Search (MNS). On a chordal graph, MLS computes a PEO (perfect elimination ordering) of the graph. We show how the algorithm MLS can be modified to com...

The secondary polytope of a point configuration A is a polytope whose face poset is isomorphic to the poset of all regular subdivisions of A. While the vertices of the secondary polytope – corresponding to the triangulations of A – are very well studied, there is not much known about the facets of the secondary polytope. The splits of a polytope, subdivisions with exactly two maximal faces, are...

Let us write f(n, ∆; C2k+1) for the maximal number of edges in a graph of order n and maximum degree ∆ that contains no cycles of length 2k + 1. For n 2 ≤ ∆ ≤ n − k − 1 and n sufficiently large we show that f(n, ∆; C2k+1) = ∆(n −∆), with the unique extremal graph a complete

A graph is clique-Helly if any family of mutually intersecting (maximal) cliques has non-empty intersection, and it is hereditary clique-Helly (abbreviated HCH) if its induced subgraphs are clique-Helly. The clique graph of a graph G is the intersection graph of its cliques, and G is self-clique if it is connected and isomorphic to its clique graph. We show that every HCH graph is an induced su...