50. For each of these graphs, find к(G), λ(G), and minyɛv deg (v), and determine which of the two inequal- ities in K (G) ≤ λ(G) ≤ minyey deg(v) are strict. a) b d

Let T:V→V be the adjacency operator of the Petersen graph as illustrated in the enclosed file. Here Vis the vector space of all formal real linear combinations of the vertices v1,..,v10 of the Petersen graph. Question: compute the spectrum of the adjacency operator Tof the Petersen graph. What do you observe?

Provide a condition that is sufficient, but not necessary, for a graph to not have an Eulerian cycle even if the graph does not include any directed edges. What are your arguments in support of your stance?

Graph Complement and Planarity,(a) Let G be a 4-regular plane graph with 10 faces. Determine how many vertices and edges G hasand give a plane drawing of such a graph.  for this problem, you must writecomplete sentences, include all details, show all of your work, and clarify all of your reasoning.

A directed edge from a vertex to itself counts. Explain each answer.

Give a sufficient but not necessary condition for a graph to be undirected and without an Eulerian cycle. Please explain your stance.

please explain fully

please explain fully

1. Give all cubic graphs you could find that have order 8, are Hamiltonian, and contain no 3-cycle. Attention: You must draw the 8-cycle first, for each of the graphs you give.

SOLVE STEP BY STEP AND IN DIGITAL FORMAT 3. Prove that every plane graph contains at least two vertices of degree less than or equal to 5.