Let G be a simple undirected graph on n vertices. V. Nikiforov studied hybrids of AG and DG defined the matrix AαG for every real α∈[0,1] as AαG=αDG+(1−α)AG. In this paper, we define generalized Randić G, introduce establish bounds Estrada index new matrix. Furthermore, find smallest value α which is positive semidefinite. Finally, present solution to problem proposed by Nikiforov. The consists...

We prove that every triangle-free planar graph on n vertices with maximum degree three has an independent set with size at least 38n. This was suggested and later conjectured by Albertson, Bollobás, and Tucker.

This article contrasts Mark Bevir’s approach to the history of ideas with a neoGramscian theory of discourse. Bevir puts the case for an ‘anti-foundationalist’ approach to understanding ideas, yet he defends a weak rationalism centred on individual intentions as the original source of all meanings. Discourse theorists specifically Ernesto Laclau and Chantal Mouffe also adopt an anti-foundationa...

The calculation of centrality measures is common practice in the study of networks, as they attempt to quantify the importance of individual vertices, edges, or other components. Different centralities attempt to measure importance in different ways. In this paper, we examine a conjecture posed by E. Estrada regarding the ability of several measures to distinguish the vertices of networks. Estr...

This work brings together ideas of mixing graph colourings, discrete homotopy, and precolouring extension. A particular focus is circular colourings. We prove that all the (k, q)-colourings of a graph G can be obtained by successively recolouring a single vertex provided k/q ≥ 2col(G) along the lines of Cereceda, van den Heuvel and Johnson’s result for kcolourings. We give various bounds for su...