Journal of Prime Research in Mathematics

Abstract: Let \(G = (V, E)\) be a simple connected graph with vertex set \(V\) and edge set \(E\). For two vertices \(u\) and \(v\) in a graph \(G\), the distance \(d(u, v)\) is the shortest path between \(u\) and \(v\) in \(G\). Graph theory has much advancements in the field of theoretical chemistry. Recently, chemical graph theory is becoming very popular among researchers because of its wide applications of mathematics in chemistry. One of the important distance based topological index is the Wiener index, defined as the sum of distances between all pairs of vertices of \(G\), defined as \(W(G) = \sum_{ u,v∈V (G)} d(u, v)\). The Hosaya polynomial is defined as \(H(G, x) =\sum _{u,v∈V (G)} x ^{d(u,v)}\). The hyper Wiener index is defined as \(WW(G) =\sum_{u,v∈V (G)} d(u, v) + \frac{1}{2}\sum_{u,v∈V (G)}d^{2}(u, v)\). In this paper, we study and compute Hosaya polynomial, Wiener index and hyper Wiener index for Jahangir graph \(J_{6,m}\), \(m ≥ 3\). Furthermore, we give exact values of these topological indices.

Abstract: Let \(R\) be a ring with identity and \(M\) be a unitary left Rmodule. The co-intersection graph of proper submodules of \(M, Ω(M)\) is an undirected simple graph whose vertices are non-trivial submodule of \(M\) in which two vertices N and K are joined by an edge, if and only if \(N + K \neq M\). In this paper, we study several properties of \(Ω(M)\). We prove that, if \(Ω(M)\) is a path, then \(Ω(M) \cong P_2 \)or \(Ω(M) \cong P_3\). We show that, if \(Ω(M)\) is a forest, then each component of \(Ω(M)\) is complete or star graph. We determine the conditions under which \(Ω(M)\) is weakly perfect. Moreover, we introduce the universal vertices and the dominating sets of \(Ω(M)\) and their relationship with the non-trivial small submodules of \(M\).

Abstract: Let \(G\) be a simple connected molecular graph in chemical graph theory, then its vertices correspond to the atoms and the edges to the bonds. Chemical graph theory is an important branch of graph theory, such that there exits many topological indices in it. Also, computing topological indices of molecular graphs is an important branch of chemical graph theory. Topological indices are numerical parameters of a molecular graph \(G\) which characterize its topology. In the present study we compute and report several results of the Narumi-Katayama and modified NarumiKatayama indices for some widely used chemical molecular structures.

Abstract: Let \(G\) be a graph with p vertices and q edges and \(A = {0, 1, 2, · · · ,\frac{q}{2}}\). A vertex labeling \(f : V (G) → A\) induces an edge labeling \(f^∗\) defined by \(f^∗ (uv) = f(u) + f(v)\) for all edges \(uv\). For \(a ∈ A\), let \(v_f (a)\) be the number of vertices \(v\) with\( f(v) = a\). A graph \(G\) is said to be vertex equitable if there exists a vertex labeling f such that for all \(a\) and \(b\) in A, \(|v_f (a) − v_f (b)| ≤ 1\) and the induced edge labels are \({1, 2, 3, · · · , q}\). In this paper, we prove that triangular ladder \(T L_n, L_n ⊙ mK_1, Q_n ⊙ K_1, T L_n⊙K_1\) and alternate triangular snake \(A(T_n)\) are vertex equitable graphs.