Laplacian eigenvalue distribution and graph parameters

Comparison of Contents Design in Mathematical Inquiry in High School Textbooks

“…If d ≥ 4, G n,d,t is the graph G * in the proof of previous theorem. As observed in[1], if Conjecture 1.1 holds, then the the upper bound given in the conjecture is sharp. The following lemma provides a family of ⌊ d 2 ⌋ (nonisomorphic) graphs for which the bound in Theorem 3.1 is achieved.Lemma 3.1. m G n,d,t [n − d + 2, n] = n − d. Proof.…”

“…For a connected graph G with u, v ∈ V (G), the distance between u and v in G is the length of a shortest path connecting them in G. The diameter of G is the greatest distance between vertices in G. After showing that if G is a connected graph of order n with diameter d ≥ 4, then m G (n − d + 3, n] ≤ n − d − 1, Ahanjideh, Akbari, Fakharan and Trevisan [1] posed the following conjecture and verified the validity for d ∈ {2, 3}.…”

Distance Laplacian eigenvalues of graphs and chromatic and independence number

“…Let G be a connected graph with n vertices and m edges, where m ≥ n. Let G * be the connected graph obtained from G by deleting an edge. Let 1) , n (k−1) , 0 . Lemma 2.5.…”

“…Some work in this direction can be seen in [5]. Recently, Ahanjideh et al [1] obtained bounds for m L(G) I in terms of structural parameters of G. In particular, they showed that…”

Distance Laplacian eigenvalues of graphs, and chromatic and independence number

“…Some work in this direction can be seen in [5]. Recently, Ahanjideh et al [1] obtained bounds for m L(G) I in terms of structural parameters of G. In particular, they showed that m L(G) ( respectively. Besides, they also proved that for a triangle-free or quadrangle-free graph G, m L(G) (n − 1, n] ≤ 1.…”