Adjacency matrix and Wiener index of zero divisor graph $\varGamma (Z_n)$

Abstract

Topological indices are widely used for characterizing molecular graphs, establishing relationships between structure and properties of molecules. In this article we discuss adjacency matrix and three topological indices: Wiener index, Laplacian energy and Zagreb indices of zero-divisor graphs of $\mathbb {Z}_n$. We also provide a MATLAB code for Laplacian energy and Zagreb indices of $\varGamma (\mathbb {Z}_n)$.

Appendix

The incidence matrix of zero-divisor graph of $\mathbb {Z}_{pqr}$ is

$$\begin{aligned} \begin{array}{c|cccccc} &{} B_1 &{} B_2 &{} B_3 &{} B_4 &{} B_5 &{} B_6 \\ \hline A_1 &{} N_1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ A_2 &{} 0 &{} N_2 &{} 0 &{} 0 &{} 0 &{} 0 \\ A_3 &{} 0 &{} 0 &{} N_3 &{} 0 &{} 0 &{} 0 \\ A_4 &{} 0 &{} 0 &{} N_4 &{} N_5 &{} N_6 &{} 0 \\ A_5 &{} 0 &{} N_7 &{} 0 &{} N_8 &{} 0 &{} N_9 \\ A_6 &{} N_{10} &{} 0 &{} 0 &{} 0 &{} N_{11} &{} N_{12} \end{array} \end{aligned}$$

where

$$\begin{aligned} B_1= & {} \{[px,qry] \mid px \in A_1, qry \in A_6\} \\ B_2= & {} \{[qx,pry] \mid qx \in A_2, pry \in A_5\} \\ B_3= & {} \{[rx,pqy] \mid rx \in A_3, pqy \in A_4\} \\ B_4= & {} \{[pqx,pry] \mid pqx \in A_4, pry \in A_5\} \\ B_5= & {} \{[pqx,qry] \mid pqx \in A_4, qry \in A_6\} \\ B_6= & {} \{[prx,qry] \mid prx \in A_5, qry \in A_6\} \end{aligned}$$

each row of matrix

• $N_1$ has entry 1 in $i + (q-1)(r-1)j^{th}$ position, $j=0,1,\ldots ,(p-2)$,

• $N_2$ has entry 1 in $i + (p-1)(r-1)j^{th}$ position, $j=0,1,\ldots ,(q-2)$,

• $N_3$ has entry 1 in $i + (p-1)(q-1)j^{th}$ position, $j=0,1,\ldots ,(r-2)$,

• $N_5$ has entry 1 in $i + (r-1)j^{th}$ position, $j=0,1,\ldots ,(q-2)$,

• $N_6$ has entry 1 in $i + (r-1)j^{th}$ position, $j=0,1,\ldots ,(p-2)$,

• $N_8$ has entry 1 in $i + (q-1)j^{th}$ position, $j=0,1,\ldots ,(r-2)$,

• $N_9$ has entry 1 in $i + (q-1)j^{th}$ position, $j=0,1,\ldots ,(p-2)$,

• $N_{11}$ has entry 1 in $i + (p-1)j^{th}$ position, $j=0,1,\ldots ,(r-2)$,

• $N_{12}$ has entry 1 in $i + (p-1)j^{th}$ position, $j=0,1,\ldots ,(q-2)$,

• $N_4$ has entry 1 from $(i-1)(p-1)(q-1) + 1$ to $i(p-1)(q-1)$ position,

• $N_7$ has entry 1 from $(i-1)(p-1)(r-1) + 1$ to $i(p-1)(r-1)$ position,

• $N_{10}$ has entry 1 from $(i-1)(q-1)(r-1) + 1$ to $i(q-1)(r-1)$ position,

here i represents the row number of each matrix and the elements of $B_i$ are ordered as: $[a_1,b_1],[a_2,b_1],\ldots ,[a_n,b_1],[a_1,b_2],[a_2,b_2],\ldots ,[a_n,b_2],\ldots ,[a_1,b_m],$

$[a_2,b_m],\ldots ,[a_n,b_m]$.