Monophonic Polynomial of the Join of Graphs

The monophonic polynomial of a graph $G$, denoted by $M(G,x)$, is the polynomial $M(G,x) = \sum_{k=m(G)}^{|G|}M(G, k)x^k $, where $|G|$ is the order of $G$ and $M(G, k)$ is the number of monophonic sets in $G$ with cardinality $k$. In this paper, we delve into some characterizations of monophonic sets in the join of two graphs and use it to determine its corresponding monophonic polynomial. Moreover, we also present the monophonic polynomials of the complete graph $K_n$ $(n \geq 1)$, the path $P_n$ $(n \geq 3)$, the cycle $C_n$ $(n \geq 4)$, the fan $F_n$ $(n \geq 3)$, the wheel $W_n$ $(n \geq 4)$, the complete bipartite $K_{m,n}$ $(m, n \geq 1)$, $P_m + P_n$ ($m, n \geq 3$), $C_m + C_n$ ($m, n \geq 4$), $P_m + C_n$ ($m \geq 3$ and $n \geq 4$), $P_m + \overline{K_n}$ ($m \geq 3$ and $n \geq 2$), and $C_m + \overline{K_n}$ ($m \geq 4$ and $n \geq 2$). In general, we obtain the monophonic polynomial of the join of two graphs.