Let \(G=(V, E)\) be a finite simple (no loops or multiple edges) undirected graph with vertex set \(V(G)=\{x_1, x_2,\dots, x_n\}\) and edge set \(E(G)\). Let \(I(G)=(x_ix_j\mid \{x_i,x_j\}\in E(G))\) be an ideal in the polynomial ring \(R=\mathbb{K}[x_1,\dots,x_n]\), where the vertex \(x_i\) is identified with the variable \(x_i\). The ideal \(I(G)\) is called the edge ideal of \(G\).
The authors study edge ideals mainly to investigate relations between algebraic properties of edge ideals and combinatorial properties of graphs. They mainly focus on describing invariants of \(I(G)\) in terms of \(G\). The organization of the paper is as follows: In Sect. 2 the authors recall some definitions and introduce the notion of the strongly disjoint set of bouquets. Moreover, in this section, they recall some well-known results by M. Hochster [in: Proc. Second Conf., Univ. Oklahoma, Norman, Okla. 171–223 (1975; Zbl 0351.13009)] and M. Katzman [J. Comb. Theory, Ser. A 113, No. 3, 435–454 (2006; Zbl 1102.13024)]. In Sect. 3 the authors introduce the crown graphs and prove some results. In particular, they prove Theorems 3.1 and 3.9. In Sect. 4 the authors discuss domination parameters of graphs and present a result on projective dimension of edge ideals of crown graphs.