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\begin{document}$$\Gamma $$\end{document} be a finite graph and let A(Γ)\documentclass[12pt]{minimal}
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\begin{document}$$A(\Gamma )$$\end{document} be its adjacency matrix. Then Γ\documentclass[12pt]{minimal}
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\begin{document}$$\Gamma $$\end{document} is singular if A(Γ)\documentclass[12pt]{minimal}
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\begin{document}$$A(\Gamma )$$\end{document} is singular. The singularity of graphs is of certain interest in graph theory and algebraic combinatorics. Here we investigate this problem for Cayley graphs Cay(G,H)\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm{Cay}(G,H)$$\end{document} when G is a finite group and when the connecting set H is a union of conjugacy classes of G. In this situation, the singularity problem reduces to finding an irreducible character χ\documentclass[12pt]{minimal}
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\begin{document}$$\chi $$\end{document} of G for which ∑h∈Hχ(h)=0.\documentclass[12pt]{minimal}
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\begin{document}$$\sum _{h\in H}\,\chi (h)=0.$$\end{document} At this stage, we focus on the case when H is a single conjugacy class hG\documentclass[12pt]{minimal}
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\begin{document}$$h^G$$\end{document} of G; in this case, the above equality is equivalent to χ(h)=0\documentclass[12pt]{minimal}
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\begin{document}$$\chi (h)=0$$\end{document}. Much is known in this situation, with essential information coming from the block theory of representations of finite groups. An element h∈G\documentclass[12pt]{minimal}
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\begin{document}$$h\in G$$\end{document} is called vanishing if χ(h)=0\documentclass[12pt]{minimal}
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\begin{document}$$\chi (h)=0$$\end{document} for some irreducible character χ\documentclass[12pt]{minimal}
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\begin{document}$$\chi $$\end{document} of G. We study vanishing elements mainly in finite simple groups and in alternating groups in particular. We suggest some approaches for constructing singular Cayley graphs.
