Year: 2012
Type: preprint
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.
Source: arXiv (Cornell University)
Author: Kranda, Dan
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W1807317062