Are there any two isomorphic graphs even though their incidence matrices are diffrent?

Question

The question is, "is it true or false? state your reasons. There are some isomorphic graphs even though their incidence matrices are different." Is it true? OR false? If it is true, could you show me some examples?

Thank you.

Answer 1

Take $A(G_1)=$ $$\begin{pmatrix}v_1:1&0&0\\v_2 :1&1&0\\v_3 :0&1&1\\v_4 :0&0&1\\ \end{pmatrix}$$ and $A(G_2)=$ $$\begin{pmatrix}v_1 :0&0&1\\v_2 :0&1&1\\v_3 :1&1&0\\v_4 :1&0&0\\ \end{pmatrix}$$ where $v_i$, $1\le i\le 4$, are the vertices and the each column vector correspond to an edge $e_j$ , $1\le j\le 3$. Obviously $A(G1)\ne$$A(G_2)$ but their corresponding graphs $G_1$ and $G_2$ are isomorphic.

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Note: Consider $e_1$,$e_2$ and $e_3$ above the top row of the matrix so that $A(G_i)$ represents an incidence matrix. I tried my best to do the same but still struggling with Mathjax on mobile.