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On the \({A_{\!\mathbb {C}}}\)-rank of multidigraphs

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Abstract

The complex adjacency matrix \({A_{\!\mathbb {C}}}(G)\) for a multidigraph G is introduced in Barik and Sahoo (AKCE Int J Graphs Comb 17(1):466–479, 2020). We study the rank of multidigraphs corresponding to the complex adjacency matrix and call it \({A_{\!\mathbb {C}}}\)-rank. It is known that a connected graph G has rank 2 if and only if G is a complete bipartite graph, and has rank 3 if and only if it is a complete tripartite graph (Cheng in Electron J Linear Algebra 16:60–67, 2007). We observe that these results hold as special cases for multidigraphs but are not sufficient. In this article, we characterize all multidigraphs with \({A_{\!\mathbb {C}}}\)-rank 2 and 3, respectively.

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Acknowledgements

The first author acknowledges SERB, DST, Government of India for the financial support (Project No.-MTR/2017/000080). The second author acknowledges the financial support from UGC, Government of India.

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  1. School of Basic Sciences, IIT Bhubaneswar, Bhubaneswar, 752050, India

    Sasmita Barik & Sane Umesh Reddy

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  1. Sasmita Barik
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  2. Sane Umesh Reddy
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Correspondence to Sasmita Barik.

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Barik, S., Reddy, S.U. On the \({A_{\!\mathbb {C}}}\)-rank of multidigraphs. Aequat. Math. 98, 189–213 (2024). https://doi.org/10.1007/s00010-023-01020-6

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