We consider finitely presented groups admitting 0-combings which are both Lipschitz (in the sense of Thurston) and tame (as defined by Mihalik and Tschantz in [8]). What we prove is that such groups are easy (and hence QSF by [11]), in the sense that they admit an easy representation (that is a map from a 2-complex to a singular 3-manifold associated to the group, satisfying several topological conditions with a strong control over singularities). Besides its own interest, one may also try to adapt the proof in a wider context, namely for groups admitting tame 1-combings (as in [8]), in order to prove the easy-representability for a larger class of finitely presented groups (note that there are still no examples of finitely presented groups which are not tame 1-combable).

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.

Closeness is a measure that quantifies how quickly information can spread from a given node to all other nodes in the network, reflecting the efficiency of communication within the network by indicating how close a node is to all other nodes. For a graph G, the subset S of vertices of $V\left(G\right)$ is called vertex cut of G if the graph $G-S$ becomes disconnected. The minimum cardinality of S for which $G-S$ is either disconnected or contains precisely one vertex is called connectivity of G. A graph is called k-connected if it stays connected even when any set of fewer than k vertices is removed. In communication networks, a k-connected graph improves network reliability; even if up to $k-1$ nodes fail, the network remains operational, maintaining connectivity between devices. This paper aims to study the concept of closeness within n-vertex graphs with fixed connectivity. First, we identify the graphs that maximize the closeness among all graphs of order n with fixed connectivity k. Then, we determine the graphs that achieve the maximum closeness within all k-connected graphs of order n, given specific fixed parameters such as diameter, independence number, and minimum degree.

Some years ago, Shitov proved that the chromatic number of the power graph of semi-groups is at most countable, thus answering a question raised by Aalipour et al. about whether this statement holds for groups. Later, Dalal and Kumar proved that the chromatic number of the enhanced power graph of a group is also countable. In the line of these recent works, we first generalize the concept of power graphs to coprime power graph, and then we prove the same type of result for this generalization. Furthermore, we state a conjecture about their common relationship as graphs.