Graphs whose edge rings have regularity two are not completely characterized. We, therefore, study the minimal graded free resolution of edge rings of certain families of graphs, such as the general barbell graph [Formula: see text] (or [Formula: see text]-barbell graph) and the complete [Formula: see text]-sunlet graph [Formula: see text], and prove that the edge ring of a barbell graph is of regularity two. We give combinatorial formulas for the graded Betti numbers in the linear strand of Stanley-Reisner rings of [Formula: see text] and [Formula: see text]. We also determine the Hilbert series of the edge ring of [Formula: see text]-barbell graph in terms of [Formula: see text]-faces of its independence complex and in terms of graded Betti numbers of the Stanley-Reisner ring of [Formula: see text]. Then we compute other graded Betti numbers of the Stanley-Reisner ring of [Formula: see text]-complexes.
Let [Formula: see text] be a graph with no isolated vertex. A dominated coloring of [Formula: see text] is a proper coloring of [Formula: see text] such that each color class is dominated by at least one vertex. The minimum number of colors needed for a dominated coloring of [Formula: see text] is called the dominated...
We study a symmetry problem for the [Formula: see text]-polynomials of edge rings of bipartite graphs. Let [Formula: see text] be a bipartite graph and write [Formula: see text]. We prove that if [Formula: see text] is pseudo-Gorenstein and [Formula: see text], then [Formula: see text] is Gorenstein. Equivalently, unde...
Yuta Hatasa· Journal of Algebra and its A...· 0 citations
Let [Formula: see text] be a simple graph on [Formula: see text] vertices and [Formula: see text] denote the corresponding binomial edge ideal in [Formula: see text], where [Formula: see text] is a field. In this paper, we provide an upper bound for the regularity of binomial edge ideals of trees. Our approach leverage...
Rajiv Kumar, Paramhans Kushwaha· Journal of Algebra and its A...· 0 citations
A [Formula: see text]-power hypergraph is obtained from a graph by adding [Formula: see text] new vertices to each edge of the graph. Let [Formula: see text] be a graph obtained by adding [Formula: see text] edges and [Formula: see text] edges to the two pendant vertices of a path of length 2, respectively. A [Formula:...
Let [Formula: see text] and [Formula: see text] respectively denote the chromatic number and clique number of a graph [Formula: see text]. In this paper, we present our contributions to two open problems on the coloring of [Formula: see text]-free graphs. A question posed by Gyárfás (1987) asks for the smallest [Formul...
C. U. Angeliya, S. Choudum, Mayamma Joseph· Asian-European Journal of Ma...· 0 citations
The eccentricity of any vertex [Formula: see text] in a connected graph [Formula: see text] is the length of the largest distance from [Formula: see text] to any other vertex in [Formula: see text]. The eccentric graph of any graph [Formula: see text], denoted by [Formula: see text], is a graph with the same vertex set...
H. Deepika, T. A. Mangam· Journal of Interconnection N...· 0 citations
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