Monophonic Polynomial of the Join of Graphs - Paluga, Rolando N. (IndoMS, 2025)
The monophonic polynomial of a graph G, denoted by M(G,x), is the polynomial M(G,x) = _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 ≥ 1), the path P_n (n ≥ 3), the cycle C_n (n ≥ 4), the fan F_n (n ≥ 3), the wheel W_n (n ≥ 4), the complete bipartite K_m,n (m, n ≥ 1), P_m + P_n (m, n ≥ 3), C_m + C_n (m, n ≥ 4), P_m + C_n (m ≥ 3 and n ≥ 4), P_m + K_n (m ≥ 3 and n ≥ 2), and C_m + K_n (m ≥ 4 and n ≥ 2). In general, we obtain the monophonic polynomial of the join of two graphs.