March 3, 2026Open Access

Some Properties of Prime Graph of Cartesian Product of the Ring

Puntos clave

The prime graph of the ring exhibits unique properties related to order and size, revealing structural insights.
Calculations indicate a distinct number of triangles within PG(Z_(p_1)×Z_(p_2)), highlighting graph complexity.
Construction of the line graph of the prime graph leads to further calculations of the order and Wiener index.
Results may indicate broader applications in algebraic graph theory and related mathematical fields.

Resumen

The prime graph of the ring R, (PG (R) ) is a graph which set of vertices consists of elements of R and two different vertices are adjacent if their product in the ring is zero. We study the prime graph of cartesian product of the ring Z_ (p₁) ×Z_ (p₂) for distinct prime numbers p₁ and p₂. We find that some properties of PG (Z_ (p₁) ×Z_ (p₂) ) such as order, size, the number of triangles, and Wiener. Further, we construct the line graph of PG (Z_ (p₁) ×Z_ (p₂) ) and calculate the order, size, and Wiener index of L (PG (Z_ (p₁) ×Z_ (p₂) ) ).

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