Let G be a finite simple graph. The line graph L (G) represents adjacencies between edges of G. We define first line simplicial complex ΔL (G) of G containing Gallai and anti-Gallai simplicial ...complexes ΔΓ (G) and ΔΓ′ (G) (respectively) as spanning subcomplexes. We establish the relation between Euler characteristics of line and Gallai simplicial complexes. We prove that the shellability of a line simplicial complex does not hold in general. We give formula for Euler characteristic of line simplicial complex associated to Jahangir graph Jm,n by presenting an algorithm.
Let ΔΓ (G) be Gallai simplicial complex defined as a generalization of Gallai graph Γ (G) such that G is a finite simple graph. The Betti numbers are topological objects which were proved to be ...invariants by Poincaré. In this note, we give formula for Betti numbers of Gallai simplicial complex associated to prism graph by coding in MATLAB. We prove that a simple graph G having n vertices of the form n = 3l + 2 or 3l + 3 is f-Gallai graph for G = 4 l (respectively G = 4 l + 1 ) a graph consisting of star graphs S2l and S 2 l ′ (respectively S2l and S2l+1) with l common vertices such that l ≥ 2.
A multigraph is a nonsimple graph which is permitted to have multiple edges, that is, edges that have the same end nodes. We introduce the concept of spanning simplicial complexes ...\(\Delta_s(\mathcal{G})\) of multigraphs \(\mathcal{G}\), which provides a generalization of spanning simplicial complexes of associated simple graphs. We give first the characterization of all spanning trees of a uni-cyclic multigraph \(\mathcal{U}_{n,m}^r\) with \(n\) edges including \(r\) multiple edges within and outside the cycle of length \(m\). Then, we determine the facet ideal \(I_\mathcal{F}(\Delta_s(\mathcal{U}_{n,m}^r))\) of spanning simplicial complex \(\Delta_s(\mathcal{U}_{n,m}^r)\) and its primary decomposition. The Euler characteristic is a well-known topological and homotopic invariant to classify surfaces. Finally, we device a formula for Euler characteristic of spanning simplicial complex \(\Delta_s(\mathcal{U}_{n,m}^r)\).
Let \(G\) be a finite simple graph. The line graph \(L(G)\) represents the adjacencies between edges of \(G\). We define first the line simplicial complex \(\Delta_L(G)\) of \(G\) containing Gallai ...and anti-Gallai simplicial complexes \(\Delta_{\Gamma}(G)\) and \(\Delta_{\Gamma'}(G)\) (respectively) as spanning subcomplexes. The study of connectedness of simplicial complexes is interesting due to various combinatorial and topological aspects. In Theorem 3.3, we prove that the line simplicial complex \(\Delta_L(G)\) is connected if and only if \(G\) is connected. In Theorem 3.4, we establish the relation between Euler characteristics of line and Gallai simplicial complexes. In Section 4, we discuss the shellability of line and anti-Gallai simplicial complexes associated to various classes of graphs.
We recall first Gallai-simplicial complex \(\Delta_{\Gamma}(G)\) associated to Gallai graph \(\Gamma(G)\) of a planar graph \(G\). The Euler characteristic is a very useful topological and homotopic ...invariant to classify surfaces. In Theorems 3.2 and 3.4, we compute Euler characteristics of Gallai-simplicial complexes associated to triangular ladder and prism graphs, respectively. Let \(G\) be a finite simple graph on \(n\) vertices of the form \(n=3l+2\) or \(3l+3\). In Theorem 4.4, we prove that \(G\) will be \(f\)-Gallai graph for the following types of constructions of \(G\). Type 1. When \(n=3l+2\). \(G=\mathbb{S}_{4l}\) is a graph consisting of two copies of star graphs \(S_{2l}\) and \(S'_{2l}\) with \(l\geq 2\) having \(l\) common vertices. Type 2. When \(n=3l+3\). \(G=\mathbb{S}_{4l+1}\) is a graph consisting of two star graphs \(S_{2l}\) and \(S_{2l+1}\) with \(l\geq 2\) having \(l\) common vertices.
Conventional flat planting is commonly used for growing wheat in Pakistan and the crop is irrigated by flood irrigation, but it leads to ineffective use of applied nitrogen owing to poor aeration and ...leaching and volatilization losses. The practice also results in greater crop lodging, lower water use efficiency, and crusting of the soil surface. In contrast, bed planting of wheat not only saves water but improves fertilizer use efficiency and grain yield. Three years of pooled data from the present study showed that wheat planting on beds and nitrogen application at 120kgha−1 produced 15.06% higher grain yield than flat planting at the same nitrogen rate. Similarly, 25.04%, 15.02%, 14.59%, and 29.83% higher nitrogen uptake, nitrogen use, and agronomic and recovery efficiencies, respectively, were recorded for bed compared to flat planting. Wheat planting on beds with a nitrogen application of 80kgha−1 gave a yield similar to that of flat planting with 120kgha−1 nitrogen. However, the economic return was 29% higher in bed planting as compared to flat planting, when nitrogen was applied at 120kgha−1.
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