arXiv subjects

Shahid Muhmood

Publications and source records attributed to Shahid Muhmood.

3 recordsLinked to original sources

arXiv2017

Spanning Simplicial Complexes of Uni-Cyclic Multigraphs

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 $Δ_s(\mathcal{G})$ 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 $I_\mathcal{F}(Δ_s(\mathcal{U}_{n,m}^r))$ of spanning simplicial complex $Δ_s(\mathcal{U}_{n,m}^r)$ 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 $Δ_s(\mathcal{U}_{n,m}^r)$.

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arXiv2017

Characterizations of Line Simplicial Complexes

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 $Δ_L(G)$ of $G$ containing Gallai and anti-Gallai simplicial complexes $Δ_Γ(G)$ and $Δ_{Γ'}(G)$ (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 $Δ_L(G)$ 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.

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arXiv2017

Topological and Algebraic Characterizations of Gallai-Simplicial Complexes

We recall first Gallai-simplicial complex $Δ_Γ(G)$ associated to Gallai graph $Γ(G)$ 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.

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