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Total graphs are Laplacian integral. (English) Zbl 1498.05158

The total graph \(T(R)\) of a commutative ring \(R\) is defined as follows. The vertices of \(T(R)\) are the elements of \(R\) and two distinct vertices \(x,y \in R\) are adjacent if and only if \(x + y\) is a zero-divisor in \(R\). The total graph, in contrast with the zero-divisor graph, engages both ring operations instead of studying just the multiplication, thus encapsulating more information about the structure of the ring. Perhaps, for this reason, total graphs have attracted attention, and graph theoretical properties have been investigated to help understand the structure of the zero-divisors in \(R\), as well as establishing algebraic properties of the ring itself according to the properties of its total graph.
In the present paper, the authors study the eigenvalues of the Laplacian matrix \(L(T(R))\) of the total graph of a finite commutative ring \(R\) with identity. They present a recursive formula for computing its eigenvalues and eigenvectors. Their main result is that the spectrum of the Laplacian \(L(T(R))\) of the total graph is integral, that is, composed only of integers.

MSC:

05C50 Graphs and linear algebra (matrices, eigenvalues, etc.)
05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
05C75 Structural characterization of families of graphs
15A18 Eigenvalues, singular values, and eigenvectors
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