ON DEG-CENTRIC JACO GRAPHS

Abstract

The deg-centric graph of a graph $G$, denoted by $G_d$, is a derived graph with the vertex set same as that of $G$, and $E\left(G_d\right)= \left\{v_i v_j: d_G\left(v_i, v_j\right) \leq \operatorname{deg}_G\left(v_i\right)\right\}$. Let $X=\left\{v_i: i=4,5,6, \ldots, n\right\}$ and $G_2=P_3 \cup \eta_{n-3}$, where $V\left(\eta_{n-3}\right)=X$. By normal consecutive stepcount for $i=3,4,5, \ldots, n-1$, do as follows: To obtain $G_i$ add the edges $v_i v_{i+1}, v_i v_{i+2}, \ldots, v_i v_{i+t}$ with $t$ as the maximum such that $\operatorname{deg}_{G_i\left(v_i\right)} \leq i$. After completion of step-count $i=n-1$, label the graph $J_n(x)$. This paper presents the coloring properties, hub number, and domination number of the deg-centric graph of the family of Jaco graphs.

Received: May 1, 2025
Revised: September 17, 2025
Accepted: September 26, 2025