On the Generalized Distance Energy of Graphs

Abstract

is the distance matrix and Tr(G) is the diagonal matrix of the node transmissions. In this paper, we extend the concept of energy to the generalized distance matrix and define the generalized distance energy E D α (G). Some new upper and lower bounds for the generalized distance energy E D α (G) of G are established based on parameters including the Wiener index W(G) and the transmission degrees. Extremal graphs attaining these bounds are identified. It is found that the complete graph has the minimum generalized distance energy among all connected graphs, while the minimum is attained by the star graph among trees of order n.

Sharp Bounds on (Generalized) Distance Energy of Graphs

“…The notion of generalized distance energy of a graph G was first motivated in Alhevaz et al [17] as the average deviation of generalized distance spectrum:…”

“…We will denote by spec(G) the generalized distance spectrum of the graph G. For some recent works on the generalized distance spectrum, we direct readers to consult the papers [8,[16][17][18][19][20].…”

“…Motivated by the definitions of E L (G) and E Q (G), Alhevaz et al [17] recently defined the generalized distance energy of G as the average deviation of generalized distance spectrum:…”

Merging the Spectral Theories of Distance Estrada and Distance Signless Laplacian Estrada Indices of Graphs

“…In addition to adjacency matrix, the energy of Laplacian, distance Laplacian, signless Laplacian as well as distance signless Laplacian has also been studied; see works [4,32,33,[35][36][37] and the references therein for more details. Recently, the authors of [38] considered a novel energy with respect to the generalized distance matrix of a graph. The generalized distance energy, denoted by E D α (G), is defined as…”

“…, where E D (G) and E Q (G) denotes, respectively, the distance energy and the distance signless Laplacian energy of a graph G. This shows that the concept of generalized distance energy of a graph G merges the theories of distance energy and the distance signless Laplacian energy of a graph G. Therefore, it will be interesting to study the quantity E D α (G) and explore some properties like the bounds, the dependence on the structure of graph G, and the dependence on the parameter α and its relation with other graph-spectrum-based invariants. The authors of [38] give some bounds for E D α (G) and have investigated its dependence on the graph topology as well as the parameter α. Our aim in this section is to explore the relationship between generalized distance Estrada index D α E(G) and generalized distance energy E D α (G) of a simple connected graph G.…”