Largest eigenvalues of the discrete p-Laplacian of trees with degree sequences.
Leaves, excesses and neighborhoods
Light classes of generalized stars in polyhedral maps on surfaces
A generalized s-star, s ≥ 1, is a tree with a root Z of degree s; all other vertices have degree ≤ 2. denotes a generalized 3-star, all three maximal paths starting in Z have exactly i+1 vertices (including Z). Let be a surface of Euler characteristic χ() ≤ 0, and m():= ⎣(5 + √49-24χ( ))/2⎦. We prove: (1) Let k ≥ 1, d ≥ m() be integers. Each polyhedral map G on with a k-path (on k vertices) contains a k-path of maximum degree ≤ d in G or a generalized s-star T, s ≤ m(), on d + 2- m() vertices...
Locally graphs.
Locally regular graphs
A graph is called locally -regular if the neighbourhood of each vertex of induces a subgraph of which is regular of degree . We study graphs which are locally -regular and simultaneously regular of degree .