2023/04/02 by Nijara Konch, Konch, Nijara, A. Bharali +3
Chemistry · Computer Science · Mathematics · #05C31 #05C50 #Combinatorics (math.CO) #FOS: Mathematics #Graph theory and applications #Matrix Theory and Algorithms #Spectral Theory (math.SP) #Synthesis and Properties of Aromatic Compounds
paper · pdf · doi:10.48550/arxiv.2304.00554
openalex publication_date 2023/04/02 · openalex created_date 2023/04/06 · openalex updated_date 2026/07/28
For a graph G, the generalized adjacency matrix Aα(G) is the convex combination of the diagonal matrix D(G) and the adjacency matrix A(G) and is defined as Aα(G)=αD(G)+(1-α) A(G) for 0≤ α≤ 1. This matrix has been found to be useful in merging the spectral theories of A(G) and the signless Laplacian matrix Q(G) of the graph G. The generalized adjacency energy or Aα-energy is the mean deviation of the Aα-eigenvalues of G and is defined as E(Aα(G))=∑i=1n|pi-(2αm)/(n)|, where pi's are Aα-eigenvalues of G. In this paper, we investigate the Aα-eigenvalues of a strongly regular graph G. We observe that Aα-spectral radius p1 satisfies δ(G)≤ p1 ≤ Δ(G), where δ(G) and Δ(G) are, respectively, the smallest and the largest degrees of G. Further, we show that the complete graph is the only graph to have exactly two distinct Aα-eigenvalues. We obtain lower and upper bounds of Aα-energy in terms of order, size and extremal degrees of G. We also discuss the extremal cases of these bounds.