By Zoran Stanić
Written for mathematicians operating with the speculation of graph spectra, this e-book explores greater than four hundred inequalities for eigenvalues of the six matrices linked to finite uncomplicated graphs: the adjacency matrix, Laplacian matrix, signless Laplacian matrix, normalized Laplacian matrix, Seidel matrix, and distance matrix. The e-book starts with a short survey of the most effects and chosen purposes to comparable subject matters, together with chemistry, physics, biology, machine technology, and keep an eye on thought. the writer then proceeds to element proofs, discussions, comparisons, examples, and workouts. each one bankruptcy ends with a short survey of extra effects. the writer additionally issues to open difficulties and offers rules for additional analyzing.
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Extra info for Inequalities for Graph Eigenvalues
In the following lemma we consider another graph perturbation preserving order and size. 29 (cf. 2] or [183, 184]) Let G(k, l) be the graph obtained from a non-trivial connected graph G by attaching at a vertex u two hanging paths whose lengths are k and l. If k ≥ l ≥ 1, then (i) (ii) (iii) (iv) (v) (vi) λ1 (G(k, l)) > λ1 (G(k + 1, l − 1)), λn (G(k, l)) ≤ λn (G(k + 1, l − 1)), μ1 (G(k, l)) ≥ μ1 (G(k + 1, l − 1)), μn−1 (G(k, l)) ≥ μn−1 (G(k + 1, l − 1)), κ1 (G(k, l)) > κ1 (G(k + 1, l − 1)), κn (G(k, l)) ≥ κ1 (G(k + 1, l − 1)).
Similarly, by , a cubic graph of diameter 4 can have at most 38 vertices. 4663, respectively. 33) gives 18 = λ13 − λ12 ≥ n − 1 − 2m = 16. 3 Other inequalities Here we give more bounds for λ1 that are mainly expressed in terms of the order, size, and (some) vertex degrees. The first is a frequently used upper bound considered independently by several authors. Subsequently, it will be compared with an upper bound of Shu and Wu. 19 (Hong et al. , Nikiforov , Zhou and Cho ) For a graph G, λ1 ≤ δ −1+ (δ + 1)2 + 4(2m − δ n) .
2 In particular, it holds whenever Δ > Δ . 35) whenever p + q ≥ Δ + 1 (where q is the number of vertices of the second largest degree) . There are three similar inequalities. The proof is left as an exercise for the reader. 26 (Feng et al. 46) and λ1 ≤ max Δ+ Δ2 + 8(di mi ) : 1≤i≤n . 47) In all cases, equality holds if and only if G is regular. We continue with two lower bounds of Kumar  obtained by using a slightly different arguments. Any principal submatrix of order two of A2 is ATi Ai ATj Ai ATi A j ATj A j , where Ai is the ith column of A.
Inequalities for Graph Eigenvalues by Zoran Stanić