http://repository.essex.ac.uk/8578/1/SNRcirculantgraphs.pdf

Smith forms for adjacency matrices of circulant graphs

We give necessary and sufficient conditions for a few classes of known circulant graphs and/or digraphs to be singular. The above graph classes are generalized to $(r,s,t)$-digraphs for non-negative integers $r,s$ and $t$, and the digraph $C_n^{i,j,k,l}$, with certain restrictions. We also obtain a necessary and sufficient condition for the digraphs $C_n^{i,j,k,l}$ to be singular. Some necessary conditions are given under which the $(r,s,t)$-digraphs are singular.

Non-singular circulant graphs and digraphs

We identify the intervals where the eigenvalues of all connected cubic circulant graphs are positive or negative Then by using the symmetry we determine the closed formula that has to be computed

The energy of all connected cubic circulant graphs

For a fixed positive integer n, let Wn be the permutation matrix corresponding to n defines a circulant graph C k. The results above are then applied to characterize its singularity, and that of its complement graph. The digraph Cr,s,t is defined as that whose adjacency matrix is circulant circ(a), where a is a vector with the first r-components equal to 1, and the next s,t and n− (r + s+ t) components equal to zero, one, and zero respectively. The singularity of this digraph (graph), under certain conditions, is also shown to depend algebraically upon these parameters. A slight generalization of these graphs are also studied.

/pdf/non-singular-circulant-graphs-and-digraphs-vguhtlus1g.pdf

A graph G is said to be singular when its adjacency matrix A = A(G) is singular, and circulant when A = A(G) is a circulant matrix. In this study, two classes of circulant graphs are studied, and conditions sufficient for these graphs to be nonsingular are established.

Leonor Aquino-ruivivar

Papers

Singular and Nonsingular Circulant Graphs