Browsing by Author "Biswas, Arindam"

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    A Cheeger Type Inequality in Finite Cayley Sum Graphs
    (Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2019) Biswas, Arindam; Saha, Jyoti Prakash
    Let G be a finite group and S be a symmetric generating set of G with |S|=d. We show that if the undirected Cayley sum graph CΣ(G,S) is an expander graph and is non-bipartite, then the spectrum of its normalised adjacency operator is bounded away from −1. We also establish an explicit lower bound for the spectrum of these graphs, namely, the non-trivial eigenvalues of the normalised adjacency operator lies in the interval (−1+h(G)4η,1−h(G)22d2], where h(G) denotes the (vertex) Cheeger constant of the d-regular graph CΣ(G,S) and η=29d8. Further, we improve upon a recently obtained bound on the non-trivial spectrum of the normalised adjacency operator of the non-bipartite Cayley graph C(G,S).
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    On a Cheeger Type Inequality in Cayley Graphs of Finite Groups
    (Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2019) Biswas, Arindam
    [nicht gut kopierbar]Let G be a finite group. It was remarked by Breuillard-Green-Guralnick-Tao that if the Cayley graph C(G,S) is an expander graph and is non-bipartite then the spectrum of the adjacency operator T is bounded away from −1. In this article we are interested in explicit bounds for the spectrum of these graphs. Specifically, we show that the non-trivial spectrum of the adjacency operator lies in the interval [−1+h(G)⁴/γ,1−h(G)²/2d²], where h(G) denotes the (vertex) Cheeger constant of the d regular graph C(G,S) with respect to a symmetric set S of generators and γ=2⁹d⁶(d+1)².
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    On Co-Minimal Pairs in Abelian Groups
    (Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2019) Biswas, Arindam; Saha, Jyoti Prakash
    A pair of non-empty subsets (W,W′) in an abelian group G is a complement pair if W+W′=G. W′ is said to be minimal to W if W+(W′∖{w′})≠G,∀w′∈W′. In general, given an arbitrary subset in a group, the existence of minimal complement(s) depends on its structure. The dual problem asks that given such a set, if it is a minimal complement to some subset. We study tightness property of complement pairs (W,W′) such that both W and W′ are minimal to each other. These are termed co-minimal pairs and we show that any non-empty finite set in an arbitrary free abelian group belongs to some co-minimal pair. We also construct infinite sets forming co-minimal pairs. Finally, we remark that a result of Kwon on the existence of minimal self-complements in Z, also holds in any abelian group.