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1. Juni 2009 · This implies that G is connected. Indeed, let G 1 be a component of G with µ (G 1 ) = µ (G), and let G 2 be the nonempty union of the remainingcomponentsofG.RemoveanedgefromG 2 ,andaddanedgebetweenG 1 andG 2 .Theresulting graph is C 4 -free with m edges, but its spectral radius is larger than µ, contradicting (4).
- Vladimir Nikiforov
- 2009
1. Mai 2013 · Since we now have an upper bound on the maximum degree of an extremal graph, we focus on e lower bound. Recall that E 0 = 1 2 q (q+ 1) 2 − q. eorem 3. Let q be an even number and let G be a C 4 -free graph on q 2 + q vertices, with maximum degree + 1 or less. Then, if e denotes the number of edges of G, we have e E 0 .
- Frank A. Firke, Peter M. Kosek, Evan D. Nash, Jason Williford
- 2013
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14. Dez. 2015 · Extremal C4-free/C5-free planar graphs. We study the topic of "extremal" planar graphs, defining exP(n, H) to be the maximum number of edges possible in a planar graph on n vertices that does not contain a given graph H as a subgraph. In particular,we examine the case when H is a small cycle,obtaining exP(n,C4) ≤ 15 7 (n − 2) for all n ≥ ...
- Chris Dowden
- 2015
18. Aug. 2023 · For a graph G, V(G) and E(G) denote the vertex set and edge set of G respectively, and \(\Delta =\Delta (G)\) denotes the maximum degree of G.For \(v\in V(G),\) \(d_G(v)\) or d(v) denotes the degree of v, \(N_G(v)\) or N(v) denotes the set of all neighbors of v in G, and \(N[v]=N(v)\cup \{v\}\).
5. Sept. 2011 · We consider the following random graph process: starting with n isolated vertices, add edges uniformly at random provided no such edge creates a copy of C 4.We show that, with probability tending to 1 as n → ∞, the final graph produced by this process has maximum degree O ((n log n) 1/3) and consequently size O (n 4/3 (log n) 1/3), which are sharp up to constants.
