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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]

Abstract Given a graph H with vertices 1 , ... , s and a set of pairwise vertex disjoint graphs G 1 , ... , G s , the vertex i of H is assigned to G i. Let G be the graph obtained from the graphs G 1 , ... , G s and the edges connecting each vertex of G i with all the vertices of G j for all edge ij of H. The graph G is called the H-join of G 1 , ... , G s. Let M (G) be a matrix on a graph G. A general result on the eigenvalues of M (G) , when the all ones vector is an eigenvector of M (G i) for i = 1 , 2 , ... , s , is given. This result is applied to obtain the distance eigenvalues, the distance Laplacian eigenvalues and as well as the distance signless Laplacian eigenvalues of G when G 1 , ... , G s are regular graphs. Finally, we introduce the notions of the distance incidence energy and distance Laplacian-energy like of a graph and we derive sharp lower bounds on these two distance energies among all the connected graphs of prescribed order in terms of the vertex connectivity. The graphs for which those bounds are attained are characterized. [ABSTRACT FROM AUTHOR]

In this paper, for a simple undirected connected graph, sharp upper bounds on the distance energy, distance Laplacian energy and distance signless Laplacian energy are obtained. The graphs attaining the corresponding upper bound are characterized. [ABSTRACT FROM AUTHOR]