Let's assume you want a network to interconnect your elements (nodes). There are many network "types" .... spanning tree; Steiner tree; hub and spoke network. A very nice network that provides a bit of redundancy (extra paths) but is still quite sparse is a "delta" connection network where every node is connected to two nearby neighbors.

"Best" refers to a particular property you are aiming to optimize. It is not clear what the objective in this particular case is (max robustness, min average distance, etc.). In any case, I may suggest the following book for a good treatment of networks:

Networks. An Introduction

Mark Newman, Oxford University Press, 2010

https://global.oup.com/academic/product/networks-9780199206650

Check also this other book (available online in the link below):

Network Science, A.L. Barabási

http://barabasi.com/networksciencebook/

Thank you Carlos. Can  You mention some networks applied in different field of Engineering?.

I guess it may depend on what kind of network it is (a distribution network, a sensor network, ...) but I'd say that scale-free networks are prevalent in many systems subject to growth and/or self-organization.

Let's assume you want a network to interconnect your elements (nodes). There are many network "types" .... spanning tree; Steiner tree; hub and spoke network. A very nice network that provides a bit of redundancy (extra paths) but is still quite sparse is a "delta" connection network where every node is connected to two nearby neighbors.

Mesh:

In mesh topology, every node has a direct point-to-point connection to every other node. Because all connections are direct, the network can handle very high-volume traffic. It is also robust because if one connection fails, the others remain intact. Security is also high since data travels along a dedicated connection.

This type of topology requires a lot of cables and is, therefore, expensive. Many of the connections are also redundant since there are several different paths for data to travel from one node to another.

The three renowned network topologies are:

1) Physical topology : it is classified into various types:

     - BUS Topology: when we use Single Cable to connect multiple Nodes.

     - star topology: it looks like a star, Using Star Network we connect various nodes with independent Cable lines. All Cable lines Centrally Connected to a HUB. In this network topology, we have to make one node as Server. To establish connectivity under this Network we have to keep Switch On the Server. If Server is off then no nodes will connect each other.

      - RING Topology : we connect nodes in a Circular fashion.

       - MESH Topology: In the mesh topology, a node stays connected to multiple nodes. The connection we establish in-between nodes is a point-to-point type. Mesh topology is two types Full Mesh and Partial Mesh.If each node is having n(n-1)/2 connections that is a Full Mesh. Any single node is less than n(n-1)/2 nodes is called partial mesh topology. Troubleshooting in mesh topology is easier.

      - TREE Topology:This topology divides the network into multiple layers. The network generally we use under LAN (Local Area Network). The middle layer of this network is called distribution layer. The highest layer is called core layer. In this topology Connection, bet-ween nodes are point-to-point. Expanding a Tree Topology is easier compare to other Topologies. Troubleshooting or error detection in tree topology is not so difficult. As there are several layers you can cross check easily.

    -HYBRID Topology: it is the same as the Internet when multiple types of network Topologies comes together.

   2) Signal Topology: it is the mapping of different nodes in a network based on the paths by which the signals pass through. Signal topology generally refers to the actual path that the signals (e.g., electromagnetic, electrical, optical, etc.) take when propagating between nodes.

  3) Logical Topology: it is  the mapping of nodes based on the paths taken by actual data in the network. It is similar to signal topology, but here the determining factor is path of data, not signals. In many cases logical topology and signal topology are used interchangeably.

Thank you all for your contributions. In cellular mobile networks, all Base stations are connected to Base station controller (BSC). All BSC are connect to Mobile switching centres (MSC). AIl MSC are interconnected by a topology. What is the nice topology to interconnect  all MSC and Public switching Telephone network PSTN?. What is  happening in Global  Telephone network technology , when we make  international (ISTD)  phone calls using mobile  phone?

I suggest the hybrid topology to be used  in this case since it combines multiple types of networks together, As per your requirement you can architect any kind of Combinations under HYBRID Topology. 

In term of the global telephone network technology , its service will be better and faster.

 Thank you  Reema for the  key answer HYBRID Topology. Hybrid Topology  is a combination of many , small and Big size Topologies. What is the preferable big topology that can interconnect all [ MSC or / and PSTN  or / and Global telephone networks]   efficiently ?

The mesh topology can also be used as a big topology  for this purpose of this interconnection

In general , there are many mesh topology. What are the best  parameters of Customer(industrialist )  choice while choosing mesh topology?

There is a huge literature on k-node connected and k-edge connected networks, network loading, network design, networks with bounded rings, etc. Try looking at past issues of the journal "Networks".

 Can one give some material talks about different topology used in VLSI   and Optoelectronic Integrated circuits?

Topology for VLSI and optoelectronic IC's is a completely different ballgame. The difficulties are related to the availability of space, the number of available layers to interconnect the components, the electrical or magnetic fields in the circuit (especially if you are working with very high frequencies), the thickness of the paths if you deal with high currents, and so on. There are software packages that do the job for you, as SPICE, for example. 

Thank you Vladimir. You have explained the difficulties and challenges in Engineering point of view. can you relate  the following ?

1. VLSI ,

2. Topology

 3. networks ,

4. Graphs ( Graph theory)  

 5. Application of  Graphs in  VLSI design.

There is an interesting pdf at http://web.mit.edu/6.173/www/currentsemester/handouts/L14-interconnect-1.pdf

Also, see ten book here:

https://books.google.cl/books?id=fMxg7Mm4QCEC&pg=PT29&lpg=PT29&dq=VLSI,+networks,+topology&source=bl&ots=ONmgg0meYf&sig=qf0ci9Pq1aGxStKJFMoHQ2PJn04&hl=en&sa=X&ved=0ahUKEwjzwPO9o5bOAhUBIZAKHbPsA-oQ6AEIUzAF#v=onepage&q=VLSI%2C%20networks%2C%20topology&f=false

Are you working on VLSI, or cellular nets, or graph theory? The approaches are in general, very different... VLSI has the difficulties I mentioned. The design of cellular networks has to take into account coverage and traffic density (cells become smaller as density increases), connectivity to the public network and connection redundancy, at least in the "backbone" network. Current backbone nets are approaching a shape similar to internet. They use IP.

Look at "Design, Deployment and Performance of 4G-LTE Networks: A Practical Approach"

By Ayman ElNashar, Mohamed El-saidny, Mahmoud Sherif.

I am working in Graph theory . looking for better networks  than Honeycomb networks and  Hexagonal networks in terms of better diameter , bijection width , and network cost.  I would like to know many Applications of networks used in different engineering.

You can use fuzzy graph or vague graph to model the uncertain real world problems.