Bounds on the Distance Energy and the Distance Estrada Index of Strongly Quotient Graphs

Abstract

The notion of strongly quotient graph (SQG) was introduced by Adiga et al. (2007). In this paper, we obtain some better results for the distance energy and the distance Estrada index of any connected strongly quotient graph (CSQG) as well as some relations between the distance Estrada index and the distance energy. We also present some bounds for the distance energy and the distance Estrada index of CSQG whose diameter does not exceed two. Additionally, we show that our results improve most of the results obtained by Güngör and Bozkurt (2009) and Zaferani (2008).

Bounds of distance Estrada index of graphs

“…It is evident that our bound is better than the lower bound in (2). Notice that our bound is incomparable to the bound in (3).…”

“…Some results relating DEE(G) to the Winer index and graph energy can be found in [3,11]. Moreover, the distance Estrada index for strongly quotient graphs and Erdős-Rényi random graphs were discussed in [2] and [19], respectively. In this note, we establish some new bounds for DEE(G) involving diameter, maximum degree, and second maximum degree.…”

Estimating the distance Estrada index

“…This graph-spectrum-based structural invariant is recently proposed in [13], and some results on its bounds can be found in [3,4,22]. If we replace in (1) the D-eigenvalues {µ i } n i=1 by the eigenvalues {λ i } n i=1 of the adjacency matrix A(G), we recover the well-researched graph descriptor Estrada index [9].…”

Further Results on Distance Estrada Index of Random Graphs

“…The distance Estrada index for strongly quotient graphs was analyzed in [4]. These results relate DE E(G) to some important graph parameters such as diameter, distance energy and the Wiener index.…”

“…This is partly inspired by the relevance of bipartite graphs in theory and various applications. Notice that some eigenvalue techniques employed in deriving bounds for distance Estrada index [4,16] or distance energy [8] require that the graph diameter does not exceed 2. Unfortunately, these methods are invalid for the study of bipartite graph since a bipartite graph of order at least 3 has diameter at least 2, and the diameter is 2 if and only if it is a complete bipartite graph.…”