Generalized distance spectral radius of some t-partitioned transmission regular graphs

Let G be an n-vertex connected graph with the distance matrix D(G) and the diagonal matrix of transmissions Tr(G). The generalized distance matrix of a connected graph G is D-alpha,D-beta(G) = alpha D(G) + beta Tr(G), where alpha not equal 0, alpha,beta is an element of R. Through this matrix one can study distance, distance Laplacian, distance signless Laplacian, and D-alpha- spectra of G in a unified way. A simple connected graph G is called t-partitioned transmission regular if the vertex set of G can be partitioned as V (G) = boolean OR V-t(i=1) (i) such that for any pair of indices i,j (not necessarily distinct) in {1, 2, . . . ,t} and x is an element of V-i, k(ij) = Sigma(y)is an element of V(j)d(x,y) is constant. In this paper, for the spider graph S-n(m) and the radar graph R-n(m), we find matrices of order m + 1 (quotient matrices) corresponding to D-alpha,D-beta(S-n(m)) and D-alpha,D-beta(R-n(m)) such that the largest eigenvalues of these matrices are generalized distance spectral radius of S-n(m) and R-n(m), respectively. Moreover, we determine the full D-alpha,D-beta-spectrum of S-n(2) and R-n(1) (wheel graph).