Let \(G = (V, E)\) be a graph. The Gutman-Milovanović index of a graph \(G\) is defined as \(\sum\limits_{uv \in E} (d(u) d(v))^{\alpha}(d(u) + d(v))^{\beta}\), where \(\alpha\) and \(\beta\) are any real numbers and \(d(u)\) and \(d(v)\) are the degrees of vertices \(u\) and \(v\) in \(G\), respectively. In this note, we present sufficient conditions based on the Gutman-Milovanović index with \(\alpha > 0\) and \(\beta >0\) for some Hamiltonian properties of a graph. We also present upper bounds for the Gutman-Milovanović index of a graph for different ranges of \(\alpha\) and \(\beta\).

Suppose \(G_1=(V_1, E_1)\) is a graph and \(G_2=(V_2, E_2)\) is a strong digraph of \(G_1\), where \(V_1\) and \(V_2\) represent the vertex sets, \(E_1\) and \(E_2\) represent the edge sets. Let \(u\) and \(v\) be any two vertices of \(G_2\). The strong distance \(sd(u,v)\) is the minimum value of edges in a strong subdiagraph of \(G_2\) that contains \(u\) and \(v\). The minimum strong diameter of \(G_2\) is defined as the maximum eccentricity \(se(u)\) from \(u\) to all other vertices in \(G_2\). In this paper, we propose different strong orientation methods to explore the minimum strong diameter of the strong product graph of \(K_{m_1,m_2,\ldots,m_k}\otimes P_n\), where \(K_{m_1,m_2,\ldots,m_k}\) and \(P_n\) represent respectively complete multipartite graph and path. ‌‌In addition, based on strong orientation methods, a new algorithm is proposed to model the presence or absence of a minimum strong diameter in a strong product graph. Simulation experiments show a trend of simultaneous decrease and concentration in the minimum strong diameter of the strong product graph, as the value of parts in \(K_{m_1,m_2,\ldots,m_k}\) increases while the length of \(P_n\) remains constant.

We consider a joint ordered multifactorisation for a given positive integer \(n\geq 2\) into \(m\) parts, where \(n=n_1~\times~\ldots~\times~n_m\), and each part \(n_j\) is split into one or more component factors. Our central result gives an enumeration formula for all such joint ordered multifactorisations \(\mathcal{N}_m(n)\). As an illustrative application, we show how each such factorisation can be used to uniquely construct and so count the number of distinct additive set systems (historically referred to as complementing set systems). These set systems under set addition generate the first \(n\) non-negative consecutive integers uniquely and, when each component set is centred about 0, exhibit algebraic invariances. For fixed integers \(n\) and \(m\), invariance properties for \(\mathcal{N}_m(n)\) are established. The formula for \(\mathcal{N}_m(n)\) is comprised of sums over associated divisor functions and the Stirling numbers of the second kind, and we conclude by deducing sum over divisor relations for our counting function \(\mathcal{N}_m(n)\). Some related integer sequences are also considered.

In this work we study the acyclic orientations of graphs. We obtain an encoding of the acyclic orientations of the complete \(p\)-partite graph with size of its parts \(n_1,n_2,\ldots,n_p\) via a vector with \(p\) symbols and length \(n=n_1+n_2+\ldots+n_p\) when the parts are fixed but not the vertices in each part. We also give a recursive way to construct all acyclic orientations of a complete multipartite graph, this construction can be done by computer easily in order \(\mathcal{O}(n)\). Furthermore, we obtain a closed formula for non-isomorphic acyclic orientations of both the complete multipartite graphs and the complete multipartite graphs with a directed spanning tree. Moreover, we obtain a closed formula for the number of acyclic orientations of a complete multipartite graph \(K_{n_1,\ldots,n_p}\) with labelled vertices. Finally, we obtain a way encode all acyclic orientations of an arbitrary graph as a permutation code. Using the codification mentioned above we obtain sharp upper and lower bounds of the number of acyclic orientations of a graph.

In this work, we defined almost neo balancing numbers and determined the general terms of them in terms of balancing and Lucas-balancing numbers. We also deduced some results on relationship with triangular, square triangular, Pell, Pell-Lucas numbers and these numbers. Further we formulate the sum of first \(n\)-terms of these numbers.