The Mostar index of a graph G is defined as the sum of absolute values of the differences between nu and nv over all edges e = uv of G, where nu(e) and nv(e) are respectively, the number of vertices of G lying closer to vertex u than to vertex v and the number of vertices of G lying closer to vertex v than to vertex u. A cactus is a graph in which any two cycles have at most one common vertex. In this paper, we determine all the n-vertex cacti with the largest Mostar index, and we give a sharp upper bound of the Mostar index for cacti of order n with k cycles, and characterize all the cacti that achieve this bound.

The Mostar index of a graph G is defined as the sum of absolute values of the differences between nu and nv over all edges uv of G, where nu and nv are respectively, the number of vertices of G lying closer to vertex u than to vertex v and the number of vertices of G lying closer to vertex v than to vertex u. We identify those trees with minimum and/or maximum Mostar index in the families of trees of order n with fixed parameters like the maximum degree, the diameter, number of pendent vertices by using graph transformations that decrease or increase the Mostar index.

A fundamental aspect of network analysis involves pinpointing nodes that hold significant positions within the network. Graph theory has emerged as a powerful mathematical tool for this purpose, and there exist numerous graph-theoretic parameters for analyzing the stability of the system. Within this framework, various graph-theoretic parameters contribute to network analysis. One such parameter used in network analysis is the so-called closeness, which serves as a structural measure to assess the efficiency of a node’s ability to interact with other nodes in the network. Mathematically, it measures the reciprocal of the sum of the shortest distances from a node to all other nodes in the network. A bipartite network is a particular type of network in which the nodes can be divided into two disjoint sets such that no two nodes within the same set are adjacent. This paper mainly studies the problem of determining the network that maximize the closeness within bipartite networks. To be more specific, we identify those networks that maximize the closeness over bipartite networks with a fixed number of nodes and one of the fixed parameters: connectivity, dissociation number, cut edges, and diameter.

For a graph G, the Mostar index of G is the sum of |nu-nv| over all edges e = uv of G, where nu denotes the number of vertices of G that have a smaller distance in G to u than to v, and analogously for nv. In this paper, we obtain a lower bound for the Mostar index on tricyclic graphs and identify those graphs that attain the lower bound.