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arXiv · 1207.3122

The Square of Adjacency Matrices

Dan Kranda
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Abstract

It can be shown that any symmetric $(0,1)$-matrix $A$ with $\tr A = 0$ can be interpreted as the adjacency matrix of a simple, finite graph. The square of an adjacency matrix $A^2=(s_{ij})$ has the property that $s_{ij}$ represents the number of walks of length two from vertex $i$ to vertex $j$. With this information, the motivating question behind this paper was to determine what conditions on a matrix $S$ are needed to have $S=A(G)^2$ for some graph $G$. Structural results imposed by the matrix $S$ include detecting bipartiteness or connectedness and counting four cycles. Row and column sums are examined as well as the problem of multiple nonisomorphic graphs with the same adjacency matrix squared.

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Dan Kranda. 2012-07-12. The Square of Adjacency Matrices. https://arxiv.org/abs/1207.3122

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