0\\
                  0 & 1 & 0 & 0
                  \end{pmatrix}$

                  

  $H:$            $\displaystyle A_H = \begin{pmatrix}0 & 1 & 1 & 0\\
                  1 & 0 & 0 & 1\\
                  1 & 0 & 0 & 1\\
                  0 & 1 & 1 & 0
                  \end{pmatrix}$

                  

                  
  ------ -- ----- -----------------------------------------------------

\

  ---------------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------
  \(a) Cartesian product $G \operatorname*{\Box}H$           

                                                             $\begin{array}{rl}A_{G \operatorname*{\Box}H} &= A_G \operatorname*{\oplus}A_H\\
                                                             &=A_G \operatorname*{\otimes}I_{|V_H|} + I_{|V_G|} \operatorname*{\otimes}A_H\\
                                                             &=\left(\raisebox{-0.48\height}{\includegraphics[scale=0.29]{fig1-CP-matrix.pdf}}\right)
                                                             \end{array}$

                                                             

  \(b) Direct (tensor) product $G \operatorname*{\times}H$   

                                                             $\begin{array}{rl}A_{G \operatorname*{\times}H} &= A_G \operatorname*{\otimes}A_H\\
                                                             &=\left(\raisebox{-0.48\height}{\includegraphics[scale=0.29]{fig1-DP-matrix.pdf}}\right)
                                                             \end{array}$

                                                             

  \(c) Strong product $G \operatorname*{\boxtimes}H$         

                                                             $\begin{array}{rl}A_{G \operatorname*{\boxtimes}H} &= A_G \operatorname*{\oplus}A_H + A_G \operatorname*{\otimes}A_H\\
                                                             &=\left(\raisebox{-0.48\height}{\includegraphics[scale=0.29]{fig1-SP-matrix.pdf}}\right)
                                                             \end{array}$
  ---------------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------

  -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                  Type                    \# of nodes                  \# of edges                                                      Average degree                                        Multiplex?
  ------------------------------------- --------------- ----------------------------------------- ------------------------------------------------------------------------------------------ ------------
                                                                                                                                                                                             

     CPMN $G \operatorname*{\Box}H$      $|V_G| |V_H|$          $|E_G||V_H| + |V_G||E_H|$                                               $\displaystyle                                           Yes
                                                                                                   \frac{2|E_G|}{|V_G|} + \frac{2|E_H|}{|V_H|} = \langle k_G \rangle + \langle k_H \rangle   
                                                                                                                                              $                                              

                                                                                                                                                                                             

    DPMN $G \operatorname*{\times}H$     $|V_G| |V_H|$                $2|E_G||E_H|$                                                     $\displaystyle                                            No
                                                                                                          \frac{4|E_G||E_H|}{|V_G| |V_H|} = \langle k_G \rangle \langle k_H \rangle          
                                                                                                                                              $                                              

                                                                                                                                                                                             

   SPMN $G \operatorname*{\boxtimes}H$   $|V_G| |V_H|$   $|E_G||V_H| + |V_G||E_H| + 2|E_G||E_H|$                                $\displaystyle\begin{array}{l}                                    No
                                                                                                       \frac{2|E_G|}{|V_G|} + \frac{2|E_H|}{|V_H|} + \frac{4|E_G||E_H|}{|V_G| |V_H|} \\      
                                                                                                    = \langle k_G \rangle + \langle k_H \rangle + \langle k_G \rangle \langle k_H \rangle    
                                                                                                                                         \end{array}$                                        

                                                                                                                                                                                             
  -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

  : Some of known topological properties of GPMNs with simple, undirected factor 