## On the distance spectrum of a cycle.(English)Zbl 0605.05029

For a graph G with n vertices the distance matrix $$D=D(G)$$ is a square matrix of order n whose elements are defined by: $$d_{rr}=0$$ and $$d_{rs}$$ $$=$$ the length of the shortest path between vertices r and s. The collection of the eigenvalues $$x_ j$$, $$j=1,2,...,n$$, of D is called the distance spectrum of G. The full treatment of the distance spectrum of a cycle $$C_ n$$ is given in the paper.
In the case of an even cycle, $$C_{2k}$$, among 2k eigenvalues of $$D(C_{2k})$$ there are k negative eigenvalues, the zero eigenvalue whose degeneracy equals (k-1), and only one positive eigenvalue which is equal to $$k^ 2$$. Among k negative eigenvalues there are [k/2] mutually distinct, doubly degenerate eigenvalues, and in addition, for k being an odd number, there is also a single negative eigenvalue which is equal to -1.
In the case of an odd cycle, $$C_{2k+1}$$, among $$2k+1$$ eigenvalues of $$D(C_{2k+1})$$ there are k mutually distinct, doubly degenerate negative eigenvalues and only one positive eigenvalue which is equal to $$k(k+1)$$. The explicit formulae for the distance spectrum of cycle are also presented.

### MSC:

 05C50 Graphs and linear algebra (matrices, eigenvalues, etc.) 05C38 Paths and cycles

### Keywords:

cycle graph; eigenvalues; distance spectrum; even cycle; odd cycle
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### References:

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