A graph is defined as � = (�(�), �(�)) Every vertex of a set � ⊆ �(�) is a member of a dominant set if it is connected to at least one other vertex in S. A dominating set of �(�) �� � minimum cardinality is known as �(�) − � the dominance number �(�) With this the domination number of the ���̈��� ����ℎ �3,� is investigated in this study. If every vertex � ∈ �(�)\� of a graph G is next to at least one vertex � ∈ � of the graph G, then that subset D is a dominant set. A dominating set �(�) �� � minimum cardinality is known as the domination number, or (G) of G. A ���̈��� ����ℎ �3,� is an even-order bipartite graph with the regularity �∆,� �� � ∆ − ������� for an even integer � ≥ 2 and the logarithmic constant n. with edges between the vertices (1, �), ��� ����� ������ (2, (� + 2� − 1) ��� (�\2)) for � = 0,1, … , ∆ − 1 and with vertices (�, �), for i = 1, 2 and 0 ≤ � ≤ �\2 − 1 , where for every �, 0 ≤ � ≤ �\2 − 1 , there is an edge between the vertices (1, j) We calculate the domination number in 4-regular ���̈��� ����ℎ �4,�in this study. If each � ∈ �(�)\� is adjacent to some vertex � ∈ �, then the subset D of the vertices in a graph G is said to be a dominant set. The lowest cardinality of a dominating set of � is known as the domination number, �(�)�� �, A set D ⊆ V (G) entails a complete prevailing. If a vertex � ∈ � is next to a vertex � ∈ �(�), then the condition is set. The lowest cardinality of a complete dominating set of �(�) is known as the total domination number, or ��(�) �� �. A ���̈��� ����ℎ �∆,� is an even-order bipartite graph with vertices(�, �), for i = 1, 2 and 0 ≤ � ≤ � 2 − 1, where for any even integer � ≥ 2��� 1 ≤ ∆≤ ⌊���2�⌋, �, 0 ≤ � ≤ � 2 − 1, and for � = 0,1, … , ∆ − 1, there is an edge connecting each vertex (2, (� + 2� − 1) ��� (�\2)) and j, 0 j n 2 1. In this study, the overall dominance number in the �3,� 3-regular ���̈��� graphs are determined.
