‎A signed graph is a pair $G_{\sigma}=(G‎, ‎\sigma)$‎, ‎where $G$ is a simple graph and $\sigma$‎, ‎called the signature‎, ‎is a function which assigns 1 or $-1$ to each edge of $G$‎. ‎We call $\sigma$ as an it orthogonal signature of $G$ if‎ $A(G_{\sigma})^2=D$‎, ‎where $A(G_{\sigma})$ is the signed adjacency matrix of $G_{\sigma}$ and $D$ is the diagonal matrix with vertex degrees on the diagonal‎. ‎We have shown that if $G$‎ has an orthogonal signature‎, ‎then $G$ should be regular‎.
‎Graphs with orthogonal signature were crucial in the proof of sensitivity conjecture by Huang in 2019‎. ‎Hunag used an orthogonal signature of hypercube in his proof‎. Alon and Zheng extended it from hypercubes to Cayley graphs of $\mathbb{Z}^{n}_2$‎. ‎Other researchers had considered signed graphs with just two distinct eigenvalues‎. ‎Note that the signed adjacency matrix of an orthogonal signature‎ ‎of a $k$-regular graph has only two distinct eigenvalues which are $\pm \sqrt{k}$‎. ‎We also note the signed adjacency matrix of any signature‎ of a $k$-regular graph has spectral radius in the interval $[ \sqrt{k},k]$‎. ‎So an orthogonal signature has the minimum spectral radius among all signatures of a regular graph‎. Orthogonal signatures have been classified for $3$-regular and 4-regular graphs‎. ‎A partial answer for 5-regular graphs is given by Stanic‎.
‎We will discuss general constructions for graphs with orthogonal‎ ‎signature and also the classification of orthogonal signatures of $k$-regular‎ ‎graphs for a fixed $k$‎, ‎specially for $k = 5‎, ‎6$‎. ‎This is a joint work with Bojan Mohar and Maxwell Levit‎.

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