  Graph is an abstract representation commonly used to model networked systems
and structure. In problems across various fields, including computer vision and
pattern recognition, and neuroscience, graphs are often brought into comparison
(a process is called \textit{graph matching}) or checked for symmetry.
Friendliness property of the associated adjacency matrices, specified by their
spectral properties, is important in deriving a convex relaxation of the
(intractable) discrete graph matching problem. In this work, we study
unfriendliness properties of symmetric graphs by studying its relation to the
underlying graph structure. It is revealed that a symmetric graph has two or
more subgraphs of the same topology, and are adjacent to the same set of
vertices. We then show that if adjacency matrices of symmetric graphs have
distinct eigenvalues then there exist eigenvectors orthogonal to the vector of
all ones, making them unfriendly. Relation of graph symmetry to
uncontrollability of multi-agent systems under agreement dynamics with one
controlled node is revisited. Examples of both synthetic and real-world graphs
are also given for illustrations.
