Consider a simple graph [Formula: see text] of order [Formula: see text] and size [Formula: see text] having Laplacian eigenvalues [Formula: see text]. Let [Formula: see text] be the sum of [Formula: see text] largest Laplacian eigenvalues of [Formula: see text]. Brouwer conjectured that [Formula: see text] for all [Formula: see text]. We obtain an upper bound for [Formula: see text] in terms of the clique number [Formula: see text], the number of vertices [Formula: see text] and the non-negative integers [Formula: see text] associated to the structure of the graph [Formula: see text]. We show that the Brouwer’s conjecture holds true for some new families of graphs. We use the same technique to prove that the Brouwer’s conjecture is true for a subclass of split graphs (It is already known that Brouwer’s conjecture holds for split graphs).
Let A(G) be the adjacency matrix and D(G) be the diagonal matrix of the vertex degrees of a simple connected graph G. Nikiforov defined the matrix A a ðGÞ of the convex combinations of D(G) and A(G) as A a ðGÞ ¼ aDðGÞ þ ð1 À aÞAðGÞ, for 0 a 1: If q 1 ! q 2 ! Á Á Á ! q n are the eigenvalues of A a ðGÞ (which we call a-adjacency eigenvalues of G), the a-adjacency energy of G is defined as E Aa ðGÞ ¼where n is the order and m is the size of G. We obtain upper and lower bounds for E Aa ðGÞ in terms of the order n, the size m and the Zagreb index Zg(G) associated to the structure of G. Further, we characterize the extremal graphs attaining these bounds.
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.
For a connected simple graph $ G $ with $ A_{\alpha} $ eigenvalues $ \rho_{1}\geq\rho_{2}\geq\dots\geq\rho_{n} $ and a real number $\beta $, let $ S_{\beta}^{\alpha}(G) =\sum\limits_{i=1}^{n}\rho_{i}^{\beta}$ be the sum of the $ \beta^{th} $ powers of the $ A_{\alpha} $ eigenvalues of graph $ G $. In this paper, we obtain various bounds for the graph invariant $ S_{\beta}^{\alpha}(G) $ in terms of different graph parameters. As a consequence, we obtain the bounds for the quantity $ IE^{A_{\alpha}}(G)= S_{\frac{1}{2}}^{\alpha}(G),$ the $ A_{\alpha} $ energy-like invariant of the graph $ G .$
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