For a metric space (X, d) and a scale parameter r ≥ 0, the Vitoris-Rips complex VR(X; r) is a simplicial complex on vertex set X, where a finite set σ ⊆ X is a simplex if and only if diameter of σ is at most r. For n ≥ 1, let In denotes the n-dimensional hypercube graph. In this paper, we show that VR(In; 3) has non trivial reduced homology only in dimensions 4 and 7. Therefore, we answer a question posed by Adamaszek and Adams recently.A (finite) simplicial complex ∆ is d-collapsible if it can be reduced to the void complex by repeatedly removing a face of size at most d that is contained in a unique maximal face of ∆. The collapsibility number of ∆ is the minimum integer d such that ∆ is d-collapsible. We show that the collapsibility number of VR(In; r) is 2 r for r ∈ {2, 3}.

For r ≥ 1, the r-independence complex of a graph G, denoted Ind r (G), is a simplicial complex whose faces are subsets A ⊆ V (G) such that each component of the induced subgraph G[A] has at most r vertices. In this article, we establish a relation between the distance r-domination number of G and (homological) connectivity of Ind r (G). We also prove that Ind r (G), for a chordal graph G, is either contractible or homotopy equivalent to a wedge of spheres. Given a wedge of spheres, we also provide a construction of a chordal graph whose r-independence complex has the homotopy type of the given wedge.

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We show that if a graph G satisfies certain conditions then the connectivity of neighbourhood complex N (G) is strictly less than the vertex connectivity of G. As an application, we give a relation between the connectivity of the neighbourhood complex and the vertex connectivity for stiff chordal graphs, and for weakly triangulated graphs satisfying certain properties. Further, we prove that for a graph G if there exists a vertex v satisfying the property that for any k-subset S of neighbours of v, there exists a vertexAs a consequence of this, we show that:(i) neighbourhood complexes of queen and king graphs are simply connected and (ii) if G is a (n + 1)-connected chordal graph which is not folded onto a clique of size n + 2, then N (G) is n-connected.

Posterior reversible encephalopathy syndrome (PRES) is a disease characterized by features of clinical and neuroimaging findings. The basics of recognizing the entity include a combination of clinical features and radiological findings in the presence of triggering factors. Clinically the patient can have various presentations like nausea, vomiting, blurring of vision, seizures, altered consciousness, transient motor deficits, or cortical blindness. Radiologically the findings can be picked up by computed tomography (CT), magnetic resonance imaging (MRI), or diffusion-weighted imaging. The usual factors that can trigger the PRES include blood pressure fluctuations, eclampsia/preeclampsia, renal failure, cytotoxic agents, and autoimmune conditions. With vigilant suspicion and prompt treatment, patients can be saved from serious complications. Here we wish to report a case of a 22-year male patient who developed PRES syndrome after undergoing an emergency splenectomy and recovered with conservative medical management for the condition. This syndrome can be managed well with an early diagnosis which needs keen suspicion and prompt treatment.