For any field F, the set of all functions f : V (G) → F whose sum on each maximal independent set is constant forms a vector space over F. In this paper, we show that the dimension can vary depending on the characteristic of the field. We also investigate the dimensions of these vector spaces and show that while some families, such as chordal graphs, have unbounded dimension, other families, su...

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, denoted WCW (G). Let B be a complete bipartite induced subgraph of ...

A weighted graph G is called well-covered if all its maximal independent sets have the same weight. Let S be an independent set of G (possibly, S = ∅). The subgraph G − N [S] is called a co-stable subgraph of G. We denote by CSub(G) the set of all co-stable subgraphs of G considered up to isomorphism. A class of weighted graphs P is called co-hereditary if it is closed under taking co-stable su...

A very well–covered graph is an unmixed without isolated vertices such that the height of its edge ideal half number vertices. We study these graphs by means Betti splittings and mapping cone constructions. show cover ideals Cohen–Macaulay are splittable. As a consequence, we compute explicitly minimal graded free resolution class prove have homological linear quotients. Finally, conjecture sam...

A graph is Zm-well-covered if all maximal independent sets have the same cardinality modulo m. Zm-well-covered graphs generalise well-covered graphs, those in which all independent sets have the same cardinality. Z2-well-covered graphs are also called parity graphs. A characterisation of cubic well-covered graphs was given by Campbell, Ellingham and Royle. Here we extend this to a characterisat...

A graph G is well-covered if every maximal independent set has the same cardinality q. Let ik(G) denote the number of independent sets of cardinality k in G. Brown, Dilcher, and Nowakowski conjectured that the independence sequence (i0(G), i1(G), . . . , iq(G)) was unimodal for any well-covered graph G with independence number q. Michael and Traves disproved this conjecture. Instead they posite...