unity – How to create a standard pbr material with shader graph?

I want to create a standard pbr material that is editable with the shader graph, but when I install the package and go the Create -> Shader tab, I only have 2 graph options: Empty Shader Graph, and Sub Graph. Isn’t there a shader graph that starts me off with a basic pbr material with color and reflections? I’m using unity 2020, and the default render pipeline. Do I need to switch to ldrp or hdrp?

Given graph G and vertices v and w can you non-deterministically walk the “least Hamiltonian path” from v to w, if it exists?

My understanding of non-deterministic algorithms is that they’re “as lucky as you want”.

…you can think of the algorithm as being able to make a guess at any point it wants, and a space alien magically guarantees it will always make the right/lucky guess.

For example, if you choose two vertices v and w, if there’s a vw path, then you can non-deterministically walk a vw path, luckily correctly guessing the vertices to walk in a vw path. Given this is true of paths in general, I wonder if the same can be done for Hamiltonian paths. I would think so, but I’m not sure.

Therefore, my first question is: if you can non-deterministically walk a vw path, can you non-deterministically walk a Hamiltonian path from v to w, if a Hamiltonian path exists? If you can non-deterministically walk a Hamiltonian path, I wonder if the same can be done for an additional constraint, what I’m calling a “least Hamiltonian path.” I would think so, but I’m not sure.

Define the least Hamiltonian path from v to w to be the Hamiltonian path such that the second vertex is the smallest it can be, then the third vertex is the smallest it can be, the fourth is the smallest, etc. Assume vertices are integers. For example, if we consider K5 the graph with vertices {1,2,3,4,5} and all possible edges, then the least Hamiltonian path for (v,w) = (2,3) is 21453. If you were at vertex 2, the next least vertex would be 1. Then, the next least vertex would 4. etc.

Therefore, my actual question is: Given graph G and vertices v and w can you non-deterministically walk the least Hamiltonian path from v to w, if it exists?

Motivation: If so, then perhaps there’s a space-efficient algorithm for non-deterministically walking a Hamiltonian path: You can keep a counter for vertices v and w, and go through each pair of vertices such that v<w, checking for the existence of a least Hamiltonian path. To ensure all vertices in G are reached, keep a counter for the number of vertices walked (this should equal N if there are N vertices). Also, ensure that each vertex in the path is unique. The least Hamiltonian path from v to w will be unique, therefore it can be walked multiple times (each time you walk through it, starting at v, you’ll non-deterministically guess the next least vertex in the least Hamiltonian path). You could use two counters to ensure no vertices are repeated – one for the vertex you’re checking to see if it’s unique (loop through the vertices in the least Hamiltonian path), and another counter for a second pass through the least Hamiltonian path. Ensure the second counter encounters the value of the first counter exactly once (use a boolean flag to keep track of this).

algorithms – Check if graph is regular given adjacency matrix

I have the following problem:

Write a function
whose input is an adjacency matrix A of a graph G. The function returns true
if G is a regular graph and false otherwise.

I understand that a graph is regular if the degrees of all the vertices are the same.

Therefore, I have written the following code which calculates the amount of degrees for each column:

def sumColumns(A):
  columns = ()
  for i in range(0, len(A(0))):
    total = 0
    for j in range(0, len(A)):
      total += A(j)(i)
    if total > 0:
      columns.append(total)
  return columns

def isRegularGraph(A):

  # Get list of degrees
  cd = sumColumns(A)

  if not len(cd) > 0 or  cd(0) != cd(len(cd) - 1):
    return False
  
  # Do comparisons from i to i - 1
  for i in range(0, len(cd) - 1):
    if cd(i) != cd(i + 1):
      return False
  return True

My question is, do I also need to check row wise if the number of degrees are the same column wise?

Is there a better algorithm to do this?

Detecting patterns in a graph network

If we create a random graph

Chop(LowerTriangularize(RandomReal({-1, 1}, {10, 10}), -1), 0.6) /.0 -> (Infinity) // MatrixForm
WeightedAdjacencyGraph(%)

enter image description here

How can we detect repeating patterns in larger graphs?

Particularly, patterns including a negative or positive parameter:

SetProperty(%, 
 EdgeStyle -> {x_ :> (PropertyValue({%, x}, 
       EdgeWeight) /. {a_?Positive -> 
        Directive(Thickness(Abs@a/400000) , Opacity(.5), Green), 
       b_?Negative -> 
        Directive(Thickness(Abs@b/400000) , Opacity(.5), Red)})})

enter image description here

The only thing that comes to mind is using the graph as an image and trying some pattern detection that way

microsoft excel – Line graph of a single Cell that is dynamically changing

I have one cell in Excel that is simply the subtraction result of two other cells. The subtraction number is dynamic i.e. it is continually changing throughout the day based on the values of the two original cells.
I want to plot a line graph that keeps track on the chart of every single number change in the subtraction cell. How do I do that.
Currently I can create a simple line chart but it plots ONLY the number currently showing in the cell. i.e when the number changes, the graph doesnt keep the previous number and so on…

graph theory – What is the number of perfect matchings on the one-dimensional skeleton of a $k$ dimensional cube?

Let $Q_k$ denote the one skeleton of the $k$-dimensional cube. How many perfect matchings are there in $Q_k$?

I honestly don’t even have a clue for this question. For $k=1$, there is trivially one, for $k=2$, there are two, for $k=3$, there are nine? Some people in my class have said that it should be $k^2$, but I have not seen any literature on this subject; wikipedia only gives a lower bound for this. It seems like if there were an answer, it would be provided somewhere on the internet. A sketch of a proof or a link would be great. Thanks.

graph theory – Can Tutte embedding be guaranteed that each face is convex?

In graph drawing and geometric graph theory, a Tutte embedding of a simple 3-vertex-connected planar graph is a crossing-free straight-line embedding with the properties that the outer face is a convex polygon and that each interior vertex is at the average (or barycenter) of its neighbors’ positions. Tutte embedding

But I don’t know if this means that such embedding when first we fix any
arbitrary outer face (may be convex)
can guarantee that each face is convex.

For example, will the following non-convex face $f_1$ appear in Tutte embedding embedding? If it exists, is there a way to make each face convex?

enter image description here

References:

graph theory – Necessary and sufficient condition for determining unique max flow

In the network flow problem, I’m trying to prove the following statement: $f’$ is an unique maximum flow if and only if the residual graph $G_{f’}$ is a directed acyclic graph.

For the forward direction, I will use contrapositive. If residual graph $G_{f’}$ has a cycle, then we can push flow along the cycle in $G_{f’}$, and augment the flow in the original graph $G$
Since sending flow or subtracting flow along cycle does not change the total flow of $f’$, so we have a different flow, say $f”$ with same flow value, which is $val(f’) =val(f”)$.

Conversely, suppose $f’$ and $f”$ are both max flow, then I want to prove the residual graph $ G_{f”}$ has a cycle. Consider $f’ -f”$ in $G_{f”}$, but I was wondering why $f’-f”$ will give a cycle in $G_{f”}$?

Can I get some help for the converse direction?