We disprove a conjecture of Estrada and Rodríguez-Velázquez [Phys. Rev. E 71, 056103 (2005)] that if a graph has identical subgraph centrality for all nodes, then the closeness and betweenness centralities are also identical for all nodes.

We give examples of vertex-transitive graphs with non-monotonic chromatic difference sequences, disproving a conjecture of Albertson and Collins on the monotonicity of the chromatic difference sequence of vertex-transitive graphs, and answering a question of Zhou on the achievability of circulants.

Chemical graph theory is currently expanding the use of topological indices to numerically encode chemical structure. The prediction characteristics provided by structure molecule a key feature these indices. concepts from are presented in brief discussion one its many applications chemistry, namely, quantitative structure-activity relationship (QSAR) studies and structure-property (QSPR) studi...

We show that every plane graph with maximum face size four whose all faces of size four are vertex-disjoint is cyclically 5-colorable. This answers a question of Albertson whether graphs drawn in the plane with all crossings independent are 5-colorable.

Let G be a simple graph on n vertices with list chromatic number χl = s. If each vertex of G is assigned a list of t colours Albertson, Grossman and Haas [1] asked how many of the vertices, λ t,s, are necessarily colourable from these lists? They conjectured that λt,s ≥ tn/s. Their work was extended by Chappell [2]. We improve the known lower bounds for λt,s. Primary AMS subject classification:...

Abstract The Estrada index of a graph/network is defined as the trace adjacency matrix exponential. It has been extended to other graph-theoretic matrices, such Laplacian, distance, Seidel adjacency, Harary, etc. Here, we describe many these extensions, including new ones, Gaussian, Mittag–Leffler and Onsager ones. More importantly, contextualize all indices in physico-mathematical frameworks w...

Entropy is essential. a measure of system’s molecular disorder or unpredictability, since work produced by organized motion. theory offers profound understanding the direction spontaneous change for many commonplace events. A formal definition random graph exists. It deals with relational data’s probabilistic and structural properties. The lower-order distribution an ensemble attributed graphs ...