Distance-regular graphs with small number of distinct distance eigenvalues

Distance spectrum and energy of graphs with small diameter

“…This observation enables us to improve a result reported in our previous work [1]. It has been shown that a bipartite distance-regular graph with diameter three has exactly three distinct D-eigenvalues if and only if it is the incidence graph of a symmetric BIBD with parameters (4s 2 , 2s 2 + s, s 2 + s).…”

“…The dual of the design (V, B) with the incidence matrix N is the design (V, B) * whose incidence matrix is N T . The adjacency matrix of the incidence graph G of the design (V, B) is defined as (1) 0 N N T 0 .…”

On the distance spectrum of minimal cages and associated distance biregular graphs

“…q 21 q 22 is a quotient matrix of the distance matrix of S(G). The characteristic polynomial of Q is x 2 − 2k 2 x − 1 2 k(k + 1) 2 = 0, and its larger root 1 2 [2k 2 + 2k(2k + 1)(k 2 + 1)] is the distance spectral radius of S(G) by Lemma 1.3. Next we take g = 5.…”

“…Equating the coefficients of k we get 2(λ 2 i − λ 2 j ) − 5(λ i − λ j ) = 0, and then λ i + λ j = 5 2 . Since k is the largest adjacency eigenvalue, p(k) is the largest distance eigenvalue [1]. Thus both λ i and λ j are different from k.…”

“…Every DR-graph of diameter d has exactly d + 1 distinct adjacency eigenvalues and at most d + 1 distinct D-eigenvalue [5]. The authors in [1] characterized some DR graphs with diameter three and four having exactly three distinct distance eigenvalues.…”