On the Ger\v{s}gorin disks of distance matrices of graphs

For a simple connected graph $G$, let $D(G)$, $Tr(G)$, $D^{L}(G)=Tr(G)-D(G)$, and $D^{Q}(G)=Tr(G)+D(G)$ be the distance matrix, the diagonal matrix of the vertex transmissions, the distance Laplacian matrix, and the distance signless Laplacian matrix of $G$, respectively. Atik and Panigrahi [2] suggested the study of the problem: Whether all eigenvalues, except the spectral radius, of $ D(G) $ and $ D^{Q}(G) $ lie in the smallest Ger\v{s}gorin disk? In this paper, we provide a negative answer by constructing an infinite family of counterexamples.

Graphs with the second signless Laplacian eigenvalue ≤ 4

We discuss the question of classifying the connected simple graphs H for which the second largest eigenvalue of the signless Laplacian Q(H) is ≤ 4. We discover that the question is inextricable linked to a knapsack problem with infinitely many allowed weights. We take the first few steps towards the general solution. We prove that this class of graphs is minor closed.

On graphs with minimal distance signless Laplacian energy

For a simple connected graph G of order n having distance signless Laplacian eigenvalues ρ 1 Q ≥ ρ 2 Q ≥ ⋯ ≥ ρ n Q \rho _1^Q \ge \rho _2^Q \ge \cdots \ge \rho _n^Q , the distance signless Laplacian energy DSLE(G) is defined as D S L E ( G ) = ∑ i = 1 n | ρ i Q - 2 W ( G ) n | DSLE\left( G \right) = \sum\nolimits_{i = 1}^n {\left| {\rho _i^Q - {{2W\left( G \right)} \over n}} \right|} where W(G) is the Weiner index of G. We show that the complete split graph has the minimum distance signless Laplacian energy among all connected graphs with given independence number. Further, we prove that the graph Kk ∨ ( Kt∪ Kn−k−t), 1 ≤ t ≤ ⌊ n - k 2 ⌋ 1 \le t \le \left\lfloor {{{n - k} \over 2}} \right\rfloor has the minimum distance signless Laplacian energy among all connected graphs with vertex connectivity k.

On Harary energy and Reciprocal distance Laplacian energies1

Let G be an graph simple, undirected, connected and unweighted graphs. The Reciprocal distance energy of a graph G is equal to the sum of the absolute values of the reciprocal distance eigenvalues. In this work, we find a lower bound for the Harary energy, reciprocal distance Laplacian energy and reciprocal distance signless Laplacian energy of a graph. Moreover, we find relationship between the Harary energy and Reciprocal distance Laplacian energies.