Algebraic graph theory is the study of the interplay between algebraic structures (both abstract as well as linear structures) and graph theory. Many concepts of abstract algebra have facilitated through the construction of graphs which are used as tools in computer science. Conversely, graph theory has also helped to characterize certain algebraic properties of abstract algebraic structures. In this survey, we highlight the rich interplay between the two topics viz groups and power graphs from groups. In the last decade, extensive contribution has been made towards the investigation of power graphs. Our main motive is to provide a complete survey on the connectedness of power graphs and proper power graphs, the Laplacian and adjacency spectrum of power graph, isomorphism, and automorphism of power graphs, characterization of power graphs in terms of groups. Apart from the survey of results, this paper also contains some new material such as the contents of Section 2 (which describes the interesting case of the power graph of the Mathieu group M11) and Section 6.1 (where conditions are discussed for the reduced power graph to be not connected). We conclude this paper by presenting a set of open problems and conjectures on power graphs.

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Miliˇcevi´c et al., in 2004, introduced topological indices known as Reformulated Zagreb indices, where they modified Zagreb indices using the edge-degree instead of vertex degree. In this paper, we present a simple approach to find the upper and lower bounds of the second reformulated Zagreb index, EM2(G), by using six graph operations/transformations. We prove that these operations significantly alter the value of reformulated Zagreb index. We apply these transformations and identify those graphs with cyclomatic number at most 3, namely trees, unicyclic, bicyclic and tricyclic graphs, which attain the upper and lower bounds of second reformulated Zagreb index for graphs.

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In the study of network optimization, finding the shortest path minimizing time/distance/cost from a source node to a destination node is one of the fundamental problems. Our focus here is to find the shortest path between any pair of nodes in a given undirected unweighted simple graph with the help of the sequence of powers of neighbourhood matrices. The authors recently introduced the concept of neighbourhood matrix as a novel representation of graphs using the neighbourhood sets of the vertices. In this article, an extension of the above work is presented by introducing a sequence of matrices, referred to as the sequence of powers of NM(G). It is denoted it by NM {l} (G) = [ {l} ij ], 1 ≤ l ≤ k(G) , where k(G) is called the iteration number, k(G) = ⌈log 2 diameter(G)⌉. As this sequence of matrices captures the distance between the nodes profoundly, we further develop the technique and present several characterizations. Based on the theoretical results, we present an algorithm to find the shortest path between any pair of nodes in a given graph. The proposed algorithm and the claims therein are formally validated through simulations on synthetic data and the real network data from Facebook. The empirical results are quite promising with our algorithm having best running time among all the existing well-known shortest path algorithms for the considered graph classes.