On eigenvalues of the reciprocal distance signless Laplacian matrix of graphs

Abstract

For a simple connected graph [Formula: see text], the reciprocal transmission [Formula: see text] of a vertex [Formula: see text] is defined as [Formula: see text] The reciprocal distance signless Laplacian (briefly RDSL) matrix of a connected graph [Formula: see text] is defined as [Formula: see text], where [Formula: see text] is the Harary matrix (reciprocal distance matrix) of [Formula: see text] and [Formula: see text] is the diagonal matrix of the vertex reciprocal transmissions in [Formula: see text]. In this paper, we first investigate the RDSL spectrum of the join and also joined union of graphs and we construct new pairs of equienergetic graphs by using the joined union of regular graphs. Then we obtain some bounds on the spectral radius of reciprocal distance signless Laplacian matrix of [Formula: see text], which is the largest eigenvalue of the matrix [Formula: see text]. We also characterize the extremal graphs.

The generalized reciprocal distance matrix of graphs

“…Thus, the matrix A α (G) is the unified way of A(G) and Q(G). In the same way, the matrix D α (G) connects D(G) to T r(G) with 1 2 Q D (G) situating in the middle of the interval. Recently, some spectral properties of D α -matrix and A α -matrix have been studied extensively and several results have been published.…”

“…It was proved [2] that, given a connected graph G of order n, the spectral radius of its reciprocal distance Laplacian matrix ρ(RL(G)) ≤ n if and only if its complement graph, denoted by G, is disconnected. The reciprocal distance signless Laplacian matrix, denoted by RQ(G) = RT (G) + RD(G), was introduced in [1]. Recently, the lower and upper bounds of the spectral radii of the reciprocal distance matrices and reciprocal distance signless Laplacian matrices of graphs were given in [7,22] and [1,12], respectively.…”

“…The reciprocal distance signless Laplacian matrix, denoted by RQ(G) = RT (G) + RD(G), was introduced in [1]. Recently, the lower and upper bounds of the spectral radii of the reciprocal distance matrices and reciprocal distance signless Laplacian matrices of graphs were given in [7,22] and [1,12], respectively. Su et al [19] determined the extremal graphs with maximal spectral radius of the RD-matrix among all graphs of order n and precise vertex connectivity, edge connectivity, chromatic number and independence number, respectively.…”

“…In [16], the matrix A α (G) = αD(G) + (1 − α)A(G) for α ∈ [0, 1], which is a convex linear combination of the degree diagonal matrix D(G) and adjacency matrix A(G), was proposed by Nikiforov. Clearly, A 0 (G) = A(G), A 1/2 (G) = 1 2 Q(G) and A 1 (G) = D(G). In 2019, to study the gradual change from D(G) to Q D (G), Cui et al [6] proposed the matrix D α (G) = αT r(G) + (1 − α)D(G) with α ∈ [0, 1].…”

“…In 2019, to study the gradual change from D(G) to Q D (G), Cui et al [6] proposed the matrix D α (G) = αT r(G) + (1 − α)D(G) with α ∈ [0, 1]. It is easy to see that D 0 (G) = D(G), D 1/2 (G) = 1 2 Q D (G) and D 1 (G) = T r(G). Thus, the matrix A α (G) is the unified way of A(G) and Q(G).…”