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In this paper, for a simple undirected connected graph G, the concept of the distance Randić matrix of G is defined and basic properties on its spectrum are derived. Moreover, the distance Randić energy of G is introduced. Bounds on this energy are obtained and for those which are sharp the extremal graphs are characterized. Finally, for α ∈ [ 0 , 1 ] , the concept of α - distance Randić matrix of G is studied and bounds on its energy are obtained. [ABSTRACT FROM AUTHOR]

For a connected graph G and α ∈ [ 0 , 1 ] , let D α (G) be the matrix D α (G) = α Tr (G) + (1 − α) D (G) , where D (G) is the distance matrix of G and Tr (G) is the diagonal matrix of its vertex transmissions. Let K m be a complete graph of order m. For n , s fixed, n > s , let G p = K s ∨ (K p ∪ K n − s − p) be the graph obtained from K s and K p ∪ K n − s − p and the edges connecting each vertex of K s with every vertex of K p ∪ K n − s − p. This paper presents some extremal results on the spectral radius of D α (G) that generalize previous results on the spectral radii of the distance matrix and distance signless Laplacian matrix. Among all connected graphs G on n vertices with a vertex/edge connectivity at most s , it is proved that 1. there exists a unique α _ ∈ (3 4 , 3 n − s 4 n − 2 s) such that if α ∈ [ 0 , α _) then the minimal spectral radius of D α (G) is uniquely attained by G = G 1 , 2. there exists a unique α ‾ ∈ (3 4 , 3 n − s 4 n − 2 s) , α ‾ ≥ α _ , such that if α ∈ (α ‾ , 1) then the minimal spectral radius of D α (G) is uniquely attained by G = G ⌊ n − s 2 ⌋ , and 3. if α = 1 then the minimal spectral radius of Tr (G) is n − 1 + ⌈ n − s 2 ⌉ and it is uniquely attained by G = G ⌊ n − s 2 ⌋. Furthermore, in terms of n and s , a tight lower bound l (n , s) of α _ and a tight upper bound u (n , s) of α ‾ are obtained. Finally, for s fixed, it is observed that lim n → ∞ ⁡ l (n , s) = lim n → ∞ ⁡ α _ = lim n → ∞ ⁡ u (n , s) = lim n → ∞ ⁡ α ‾ = 3 4. [ABSTRACT FROM AUTHOR]

Let G be a simple undirected connected graph. In this paper, new upper bounds on the distance Laplacian spectral radius of G are obtained. Moreover, new lower and upper hounds for the distance signless Laplacian spectral radius of G are derived. Some of the above mentioned bounds are sharp and the graphs attaining the corresponding bound are characterized. Several illustrative examples are included. [ABSTRACT FROM AUTHOR]

Let G be a simple undirected connected graph. Let D (G) be the distance matrix of G and let Tr (G) be the diagonal matrix of the vertex transmissions in G. Let α ∈ [ 0 , 1 ]. In S-Y. Cui et al. (2019) [7] the matrix D α (G) = α Tr (G) + (1 − α) D (G) is introduced and several properties are obtained. In this paper, new properties on the D α -matrix are derived including inequalities that involve the largest vertex transmission and the spectral radii of the distance matrix, distance signless Laplacian matrix and D α -matrix. The necessary and sufficient condition for the equality in each of the inequalities is given. Moreover, some results on the D α -matrix of a graph with independent sets of vertices sharing the same set of neighbors, including the case of a complete multipartite graph, are obtained. Finally, the spectrum of D α (G) is determined when G is the H -join of regular graphs. [ABSTRACT FROM AUTHOR]