Computational graph for general feynman diagrams

mutable struct FeynmanGraph{F<:Number,W}

Computational graph representation of a (collection of) Feynman diagram(s). All Feynman diagrams should share the same set of external and internal vertices.

Members:

• id::Int the unique hash id to identify the diagram
• name::Symbol name of the diagram
• orders::Vector{Int} orders associated with the Feynman graph, e.g., loop/derivative orders
• properties::FeynmanProperties diagrammatic properties, e.g., the operator vertices and topology
• subgraphs::Vector{FeynmanGraph{F,W}} vector of sub-diagrams
• subgraph_factors::Vector{F} scalar multiplicative factors associated with each subdiagram
• operator::DataType node operation (Sum, Prod, etc.)
• weight::W weight of the diagram

Example:

julia> g1 = FeynmanGraph([]; vertices=[𝑓⁺(1),𝑓⁻(2)], external_indices=[1,2], external_legs=[true,true])
1:f⁺(1)|f⁻(2)=0.0

julia> g2 = FeynmanGraph([]; vertices=[𝑓⁺(3),𝑓⁻(4)], external_indices=[1,2], external_legs=[true,true])
2:f⁺(3)|f⁻(4)=0.0

julia> g = FeynmanGraph([g1,g2]; vertices=[𝑓⁺(1),𝑓⁻(2),𝑓⁺(3),𝑓⁻(4)], operator=ComputationalGraphs.Prod(), external_indices=[1,2,3,4], external_legs=[true,true,true,true])
3:f⁺(1)|f⁻(2)|f⁺(3)|f⁻(4)=0.0=Ⓧ (1,2)

mutable struct Graph{F<:Number,W}

A representation of a computational graph, e.g., an expression tree, with type stable node data.

Members:

• id::Int the unique hash id to identify the diagram
• name::Symbol name of the diagram
• orders::Vector{Int} orders associated with the graph, e.g., derivative orders
• subgraphs::Vector{Graph{F,W}} vector of sub-diagrams
• subgraph_factors::Vector{F} scalar multiplicative factors associated with each subgraph. Note that the subgraph factors may be manipulated algebraically. To associate a fixed multiplicative factor with this graph which carries some semantic meaning, use the factor argument instead.
• operator::DataType node operation. Addition and multiplication are natively supported via operators Sum and Prod, respectively. Should be a concrete subtype of AbstractOperator.
• weight::W the weight of this node
• properties::Any extra information of Green's functions.

Example:

julia> g1 = Graph([])
1=0.0

julia> g2 = Graph([]; factor=2)
2⋅2.0=0.0

julia> g = Graph([g1, g2]; operator=ComputationalGraphs.Sum())
3=0.0=⨁ (1,2)

function Base.:*(c1, g2::Graph{F,W}) where {F,W}

Returns a graph representing the scalar multiplication c1*g2.

Arguments:

• c1 scalar multiple
• g2 Feynman graph

function Base.:*(c1, g2::Graph{F,W}) where {F,W}

Returns a graph representing the scalar multiplication c1*g2.

Arguments:

• c1 scalar multiple
• g2 computational graph

function Base.:*(g1::Graph{F,W}, c2) where {F,W}

Returns a graph representing the scalar multiplication g1*c2.

Arguments:

• g1 Feynman graph
• c2 scalar multiple

function Base.:*(g1::Graph{F,W}, c2) where {F,W}

Returns a graph representing the scalar multiplication g1*c2.

Arguments:

• g1 computational graph
• c2 scalar multiple

function Base.:*(g1::Graph{F,W}, g2::Graph{F,W}) where {F,W}

Returns a graph g1 * g2 representing the graph product between g1 and g2.

Arguments:

• g1 first computational graph
• g2 second computational graph

function Base.:+(g1::Graph{F,W}, g2::Graph{F,W}) where {F,W}

Returns a graph g1 + g2 representing the addition of two Feynman diagrams g2 with g1. Diagrams g1 and g2 must have the same diagram type, orders, and external vertices.

Arguments:

• g1 first Feynman graph
• g2 second Feynman graph

function Base.:+(g1::Graph{F,W}, g2::Graph{F,W}) where {F,W}

Returns a graph g1 + g2 representing the addition of g2 with g1. Graphs g1 and g2 must have the same orders.

Arguments:

• g1 first computational graph
• g2 second computational graph

function Base.:-(g1::Graph{F,W}, g2::Graph{F,W}) where {F,W}

Returns a graph g1 - g2 representing the subtraction of g2 from g1. Diagrams g1 and g2 must have the same diagram type, orders, and external vertices.

Arguments:

• g1 first Feynman graph
• g2 second Feynman graph

function Base.:-(g1::Graph{F,W}, g2::Graph{F,W}) where {F,W}

Returns a graph g1 - g2 representing the subtraction of g2 from g1. Graphs g1 and g2 must have the same orders.

Arguments:

• g1 first computational graph
• g2 second computational graph

function Base.convert(::Type{G}, g::FeynmanGraph{F,W}) where {F,W,G<:Graph}

Converts a FeynmanGraph `g` into a Graph, discarding its Feynman properties.
After conversion, graph `g` is no longer guaranteed to be a valid (group of) Feynman diagram(s).

# Arguments:
- `g`  computational graph

print(io::IO, graph::AbstractGraph)

Write an un-decorated text representation of an AbstractGraph `graph` to the output stream `io`.

show(io::IO, graph::AbstractGraph; kwargs...)

Write a text representation of an AbstractGraph `graph` to the output stream `io`.

Checks that all elements of an iterable x are equal.

function build_derivative_graph(graphs::AbstractVector{G}, orders::NTuple{N,Int};
    nodes_id=nothing) where {G<:Graph,N}

Constructs a derivative graph using forward automatic differentiation with given graphs and derivative orders.

Arguments:

• graphs: A vector of graphs.
• orders::NTuple{N,Int}: A tuple indicating the orders of differentiation. N represents the number of independent variables to be differentiated.
• nodes_id: Optional node IDs to indicate saving their derivative graph.

Returns:

• A dictionary containing the dual derivative graphs for all indicated nodes.

If isnothing(nodes_id), indicated nodes include all leaf and root nodes. Otherwise, indicated nodes include all root nodes and other nodes from nodes_id.

function burn_from_targetleaves!(graphs::AbstractVector{G}, targetleaves_id::AbstractVector{Int}; verbose=0) where {G <: AbstractGraph}

Removes all nodes connected to the target leaves in-place via "Prod" operators.

Arguments:

• graphs: A vector of graphs.
• targetleaves_id::AbstractVector{Int}: Vector of target leafs' id.
• verbose: Level of verbosity (default: 0).

Returns:

• The id of a constant graph with a zero factor if any graph in graphs was completely burnt; otherwise, nothing.

function collect_labels(g::FeynmanGraph)

Returns the list of sorted unique labels in graph `g`.

Arguments:

• g::FeynmanGraph: graph to find labels for

function constant_graph(factor=one(_dtype.factor))

Returns a graph that represents a constant equal to f, where f is the factor with default value 1.

Arguments:

• f: constant factor

function count_expanded_operation(g::G) where {G<:AbstractGraph}

Returns the total number of operations in the totally expanded version (without any parentheses in the mathematical expression) of the graph.

Arguments:

• g::Graph: graph for which to find the total number of operations in its expanded version.

function count_leaves(g::G) where {G<:AbstractGraph}

Returns the total number of leaves with unique id in the graph.

Arguments:

• g::Graph: graph for which to find the total number of leaves.

function count_operation(g::G) where {G<:AbstractGraph}

Returns the total number of  additions and multiplications in the graph.

Arguments:

• g::Graph: graph for which to find the total number of operations.

function diagram_type(g::FeynmanGraph)

Returns the diagram type (::DiagramType) of FeynmanGraph g.

function disconnect_subgraphs!(g::G) where {G<:AbstractGraph}

Empty the subgraphs and subgraph_factors of graph `g`. Any child nodes of g
not referenced elsewhere in the full computational graph are effectively deleted.

function drop_topology(p::FeynmanProperties)

Returns a copy of the given FeynmanProperties `p` modified to have no topology.

function eldest(g::AbstractGraph)

Returns the first child (subgraph) of a graph g.

Arguments:

• g::AbstractGraph: graph for which to find the first child

function external_indices(g::FeynmanGraph)

Returns a list of indices (::Vector{Int}}) to the external vertices of the FeynmanGraph g.

function external_labels(g::FeynmanGraph)

Returns the labels of all physical external vertices of FeynmanGraph g.

function external_legs(g::FeynmanGraph)

Returns a list of Boolean indices external_legs (::Vector{Bool}) indicating which external vertices of FeynmanGraph g have real legs (true: real leg, false: fake leg).

function external_operators(g::FeynmanGraph)

Returns all physical external operators (::OperatorProduct}) of FeynmanGraph g.

function external_vertex(ops::OperatorProduct;
    name="", factor=one(_dtype.factor), weight=zero(_dtype.weight), operator=Sum())

Create a ExternalVertex-type FeynmanGraph from given OperatorProduct ops, including several quantum operators for an purely external vertex.

function feynman_diagram(subgraphs::Vector{FeynmanGraph{F,W}}, topology::Vector{Vector{Int}}, perm_noleg::Union{Vector{Int},Nothing}=nothing;
    factor=one(_dtype.factor), weight=zero(_dtype.weight), name="", diagtype::DiagramType=GenericDiag()) where {F,W}

Create a FeynmanGraph representing feynman diagram from all subgraphs and topology (connections between vertices), where each ExternalVertex is given in vertices, while internal vertices are constructed with external legs of graphs in vertices, or simply OperatorProduct in vertices.

Arguments:

• subgraphs::Vector{FeynmanGraph{F,W}} all subgraphs of the diagram. All external operators of subgraphs constitute all operators of the new diagram.
• topology::Vector{Vector{Int}} topology of the diagram. Each Vector{Int} stores operators' index connected with each other (as a propagator).
• perm_noleg::Union{Vector{Int},Nothing}=nothing permutation of all the nonleg external operators. By default, setting nothing means to use the default order from subgraphs.
• factor::F overall scalar multiplicative factor for this diagram (e.g., permutation sign)
• weight weight of the diagram
• name name of the diagram
• diagtype type of the diagram

Example:

julia> V = [𝑓⁺(1)𝑓⁻(2)𝜙(3), 𝑓⁺(4)𝑓⁻(5)𝜙(6), 𝑓⁺(7)𝑓⁻(8)𝜙(9)];
julia> g = feynman_diagram(interaction.(V), [[1, 5], [3, 9], [4, 8]], [3, 1, 2])
7:f⁺(1)f⁻(2)ϕ(3)|f⁺(4)f⁻(5)ϕ(6)|f⁺(7)f⁻(8)ϕ(9)=0.0=Ⓧ (1,2,3,4,5,6)

julia> g.subgraphs
6-element Vector{FeynmanGraph{Float64, Float64}}:
 1:f⁺(1)f⁻(2)ϕ(3)=0.0
 2:f⁺(4)f⁻(5)ϕ(6)=0.0
 3:f⁺(7)f⁻(8)ϕ(9)=0.0
 4:f⁺(1)|f⁻(5)⋅-1.0=0.0
 5:ϕ(3)|ϕ(9)=0.0
 6:f⁺(4)|f⁻(8)⋅-1.0=0.0

function flatten_all_chains!(g::AbstractGraph; verbose=0)

F Flattens all nodes representing trivial unary chains in-place in the given graph g.

Arguments:

• graphs: The graph to be processed.
• verbose: Level of verbosity (default: 0).

Returns:

• The mutated graph g with all chains flattened.

function flatten_all_chains!(graphs::Union{Tuple,AbstractVector{<:AbstractGraph}}; verbose=0)

Flattens all nodes representing trivial unary chains in-place in the given graphs.

Arguments:

• graphs: A collection of graphs to be processed.
• verbose: Level of verbosity (default: 0).

Returns:

• The mutated collection graphs with all chains in each graph flattened.

function flatten_chains!(g::AbstractGraph)

Recursively flattens chains of subgraphs within the given graph `g` by merging certain trivial unary subgraphs 
into their parent graphs in the in-place form.

Acts only on subgraphs of `g` with the following structure: 𝓞 --- 𝓞' --- ⋯ --- 𝓞'' ⋯ (!),
where the stop-case (!) represents a leaf, a non-trivial unary operator 𝓞'''(g) != g, or a non-unary operation.

Arguments:

• g::AbstractGraph: graph to be modified

function flatten_chains(g::AbstractGraph) 

Recursively flattens chains of subgraphs within a given graph `g` by merging certain trivial unary subgraphs into their parent graphs,
This function returns a new graph with flatten chains, derived from the input graph `g` remaining unchanged.

Arguments:

• g::AbstractGraph: graph to be modified

flatten_prod!(graph::G; map::Dict{Int,G}=Dict{Int,G}()) where {G<:AbstractGraph}

Recursively merge multi-product sub-branches within the given graph `g  by merging  product subgraphs 
into their parent product graphs in the in-place form.

Arguments:

• g::AbstractGraph: graph to be modified
• map::Dict{Int,G}=Dict{Int,G}(): A dictionary that maps the id of an original node with its corresponding new node after transformation.

In recursive transform, nodes can be visited several times by different parents. This map keeps track of those visited, and reuse those transformed sub-branches instead of recreating them.

flatten_sum!(graph::G; map::Dict{Int,G}=Dict{Int,G}()) where {G<:AbstractGraph}

Recursively merge multi-product sub-branches within the given graph `g  by merging  sum subgraphs 
into their parent sum graphs in the in-place form.

Arguments:

• g::AbstractGraph: graph to be modified
• map::Dict{Int,G}=Dict{Int,G}(): A dictionary that maps the id of an original node with its corresponding new node after transformation.

In recursive transform, nodes can be visited several times by different parents. This map keeps track of those visited, and reuse those transformed sub-branches instead of recreating them.

function forwardAD(diag::Graph{F,W}, ID::Int) where {F,W}

Given a  graph G and an id of graph g, calculate the derivative d G / d g by forward propagation algorithm.

Arguments:

• diag::Graph{F,W}: Graph G to be differentiated
• ID::Int: ID of the graph g

function forwardAD_root!(graphs::AbstractVector{G}, idx::Int=1,
    dual::Dict{Tuple{Int,NTuple{N,Bool}},G}=Dict{Tuple{Int,Tuple{Bool}},G}()) where {G<:Graph,N}

Computes the forward automatic differentiation (AD) of the given graphs beginning from the roots.

Arguments:

• graphs: A vector of graphs.
• idx: Index for differentiation (default: 1).
• dual: A dictionary that holds the result of differentiation.

Returns:

• The dual dictionary populated with all differentiated graphs, including the intermediate AD.

function group(gv::AbstractVector{G}, indices::Vector{Int}) where {G<:FeynmanGraph}

Group the graphs in gv by the external operators at the indices indices. Return a dictionary of Vector{OperatorProduct} to GraphVector.

Example

julia> p1 = propagator(𝑓⁺(1)𝑓⁻(2));

julia> p2 = propagator(𝑓⁺(1)𝑓⁻(3));

julia> p3 = propagator(𝑓⁺(2)𝑓⁻(3));

julia> gv = [p1, p2, p3];

julia> ComputationalGraphs.group(gv, [1, 2])
Dict{Vector{OperatorProduct}, Vector{FeynmanGraph{Float64, Float64}}} with 3 entries:
  [f⁻(2), f⁺(1)] => [1:f⁺(1)|f⁻(2)⋅-1.0=0.0]
  [f⁻(3), f⁺(1)] => [2:f⁺(1)|f⁻(3)⋅-1.0=0.0]
  [f⁻(3), f⁺(2)] => [3:f⁺(2)|f⁻(3)⋅-1.0=0.0]

julia> ComputationalGraphs.group(gv, [1, ])
Dict{Vector{OperatorProduct}, Vector{FeynmanGraph{Float64, Float64}}} with 2 entries:
  [f⁻(3)] => [2:f⁺(1)|f⁻(3)⋅-1.0=0.0, 3:f⁺(2)|f⁻(3)⋅-1.0=0.0]
  [f⁻(2)] => [1:f⁺(1)|f⁻(2)⋅-1.0=0.0]

julia> ComputationalGraphs.group(gv, [2, ])
Dict{Vector{OperatorProduct}, Vector{FeynmanGraph{Float64, Float64}}} with 2 entries:
  [f⁺(2)] => [3:f⁺(2)|f⁻(3)⋅-1.0=0.0]
  [f⁺(1)] => [1:f⁺(1)|f⁻(2)⋅-1.0=0.0, 2:f⁺(1)|f⁻(3)⋅-1.0=0.0]

function has_zero_subfactors(g::AbstractGraph, operator_type::Type{<:AbstractOperator})

Determines whether the graph `g` has only zero-valued subgraph factors based on the specified operator type.
This function does not recurse through the subgraphs of `g`, so it only checks the immediate subgraph factors.
If `g` is a leaf (i.e., has no subgraphs), the function returns `false` by convention.

The behavior of the function depends on the operator type:
- `Sum`: Checks if all subgraph factors are zero.
- `Prod`: Checks if any subgraph factor is zero.
- `Power{N}`: Checks if the first subgraph factor is zero.
- Other `AbstractOperator`: Defaults to return `false`.

Arguments:

• g::AbstractGraph: graph to be analyzed
• operator: the operator used in graph g

function haschildren(g::AbstractGraph)

Returns whether the graph has any children (subgraphs).

Arguments:

• g::AbstractGraph: graph to be analyzed

function id(g::AbstractGraph)

Returns the unique hash id of computational graph `g`.

function interaction(ops::OperatorProduct; name="", reorder::Union{Function,Nothing}=nothing,
    factor=one(_dtype.factor), weight=zero(_dtype.weight), operator=Sum())

Create a Interaction-type FeynmanGraph from given OperatorProduct ops, including several quantum operators for a vertex. One can call a reorder function for the operators ordering.

function isexternaloperators(g::FeynmanGraph, i)

Check if i in the external indices of FeynmanGraph g.

function is_internal(g::FeynmanGraph, i)

Check if i in the internal indices of FeynmanGraph g.

function isapprox_one(x)

Returns true if x ≈ one(x).

function isassociative(g::AbstractGraph)

Returns true if the graph operation of node `g` is associative.
Otherwise, returns false.

function isbranch(g::AbstractGraph)

Returns whether the graph g is a branch-type (depth-1 and one-child) graph.

Arguments:

• g::AbstractGraph: graph to be analyzed

function ischain(g::AbstractGraph)

Returns whether the graph g is a chain-type graph (i.e., a unary string).

Arguments:

• g::AbstractGraph: graph to be analyzed

function isequiv(a::AbstractGraph, b::AbstractGraph, args...)

Determine whether graph `a` is equivalent to graph `b` without considering fields in `args`.

function isleaf(g::AbstractGraph)

Returns whether the graph g is a leaf (terminating tree node).

Arguments:

• g::AbstractGraph: graph to be analyzed

function linear_combination(graphs::Vector{FeynmanGraph{F,W}}, constants::AbstractVector=ones(F, length(graphs))) where {F,W}

Given a vector 𝐠 of graphs each with the same type and external/internal vertices and an equally-sized vector 𝐜 of constants, returns a new graph representing the linear combination (𝐜 ⋅ 𝐠). The function identifies unique graphs from the input graphs and sums their associated constants. All input Graphs must have the same diagram type, orders, and external vertices.

Arguments:

• graphs vector of input FeymanGraphs
• constants vector of scalar multiples (defaults to ones(F, length(graphs))).

Returns:

• A new FeynmanGraph{F,W} object representing the linear combination of the unique input graphs weighted by the constants,

where duplicate graphs in the input graphs are combined by summing their associated constants.

Example:

Given graphs `g1`, `g2`, `g1` and constants `c1`, `c2`, `c3`, the function computes `(c1+c3)*g1 + c2*g2`.

function linear_combination(graphs::Vector{Graph{F,W}}, constants::AbstractVector=ones(F, length(graphs))) where {F,W}

Given a vector 𝐠 of graphs and an equally-sized vector 𝐜 of constants, returns a new graph representing the linear combination (𝐜 ⋅ 𝐠). The function identifies unique graphs from the input graphs and sums their associated constants. All input graphs must have the same orders.

Arguments:

• graphs vector of computational graphs
• constants vector of scalar multiples (defaults to ones(F, length(graphs))).

Returns:

• A new Graph{F,W} object representing the linear combination of the unique input graphs weighted by the constants,

where duplicate graphs in the input graphs are combined by summing their associated constants.

Example:

Given graphs `g1`, `g2`, `g1` and constants `c1`, `c2`, `c3`, the function computes `(c1+c3)*g1 + c2*g2`.

function linear_combination(g1::FeynmanGraph{F,W}, g2::FeynmanGraph{F,W}, c1, c2) where {F,W}

Returns a graph representing the linear combination c1*g1 + c2*g2. If g1 == g2, it will return a graph representing (c1+c2)*g1 Feynman Graphs g1 and g2 must have the same diagram type, orders, and external vertices.

Arguments:

• g1 first Feynman graph
• g2 second Feynman graph
• c1: first scalar multiple (defaults to 1).
• c2: second scalar multiple (defaults to 1).

function linear_combination(g1::Graph{F,W}, g2::Graph{F,W}, c1, c2) where {F,W}

Returns a graph representing the linear combination c1*g1 + c2*g2. If g1 == g2, it will return a graph representing (c1+c2)*g1. Graphs g1 and g2 must have the same orders.

Arguments:

• g1 first computational graph
• g2 second computational graph
• c1 first scalar multiple
• c2 second scalar multiple

function mask_zero_subgraph_factors(operator::Type{<:AbstractOperator}, subg_fac::Vector{F}) where {F}

Returns a list of indices that should be considered when performing the operation (e.g., Sum, Prod, Power), effectively masking out zero values as appropriate.

The behavior of the function depends on the operator type:
- `Sum`: Returns all indices that are not equal to zero.
- `Prod`: Returns the index of the first zero value, or all indices if none are found.
- `Power`: Returns `[1]`, or error if the power is negative.
- Other `AbstractOperator`: Defaults to return all indices.

function merge_all_linear_combinations!(g::AbstractGraph; verbose=0)

Merges all nodes representing a linear combination of a non-unique list of subgraphs in-place in the given graph `g`.

Arguments:

• g: An AbstractGraph.
• verbose: Level of verbosity (default: 0).

Returns:

• Optimized graph.

function merge_all_linear_combinations!(graphs::Union{Tuple,AbstractVector{<:AbstractGraph}}; verbose=0)

Merges all nodes representing a linear combination of a non-unique list of subgraphs in-place in the given graphs.

Arguments:

• graphs: A collection of graphs to be processed.
• verbose: Level of verbosity (default: 0).

Returns:

• Optimized graphs.

function merge_all_multi_products!(g::Graph; verbose=0)

Merges all nodes representing a multi product of a non-unique list of subgraphs in-place in the given graph `g`.

Arguments:

• g::Graph: A Graph.
• verbose: Level of verbosity (default: 0).

Returns:

• Optimized graph.

function merge_all_multi_products!(graphs::Union{Tuple,AbstractVector{<:Graph}}; verbose=0)

Merges all nodes representing a multi product of a non-unique list of subgraphs in-place in the given graphs.

Arguments:

• graphs: A collection of graphs to be processed.
• verbose: Level of verbosity (default: 0).

Returns:

• Optimized graphs.

function merge_linear_combination!(g::AbstractGraph)

Modifies a computational graph `g` by factorizing multiplicative prefactors, e.g.,
3*g1 + 5*g2 + 7*g1 + 9*g2 ↦ 10*g1 + 14*g2 = linear_combination(g1, g2, 10, 14).
Returns a linear combination of unique subgraphs and their total prefactors. 
Does nothing if the graph `g` does not represent a Sum operation.

Arguments:

• g::AbstractGraph: graph to be modified

function merge_linear_combination(g::AbstractGraph)

Returns a copy of computational graph `g` with multiplicative prefactors factorized,
e.g., 3*g1 + 5*g2 + 7*g1 + 9*g2 ↦ 10*g1 + 14*g2 = linear_combination(g1, g2, 10, 14).
Returns a linear combination of unique subgraphs and their total prefactors. 
Does nothing if the graph `g` does not represent a Sum operation.

Arguments:

• g::AbstractGraph: graph to be modified

function merge_multi_product!(g::Graph{F,W}) where {F,W}

Merge multiple products within a computational graph `g` if they share the same operator (`Prod`).
If `g.operator == Prod`, this function will merge `N` identical subgraphs into a single subgraph with a power operator `Power(N)`. 
The function ensures each unique subgraph is counted and merged appropriately, preserving any distinct subgraph_factors associated with them.

Arguments:

• g::Graph: graph to be modified

Returns

• A merged computational graph with potentially fewer subgraphs if there were repeating subgraphs with the Prod operator. If the input graph's operator isn't Prod, the function returns the input graph unchanged.

function merge_multi_product(g::Graph{F,W}) where {F,W}

Returns a copy of computational graph `g` with multiple products merged if they share the same operator (`Prod`).
If `g.operator == Prod`, this function will merge `N` identical subgraphs into a single subgraph with a power operator `Power(N)`. 
The function ensures each unique subgraph is counted and merged appropriately, preserving any distinct subgraph_factors associated with them.

Arguments:

• g::Graph: graph to be modified

Returns

• A merged computational graph with potentially fewer subgraphs if there were repeating subgraphs with the Prod operator. If the input graph's operator isn't Prod, the function returns the input graph unchanged.

multi_product(graphs::Vector{Graph{F,W}}, constants::AbstractVector=ones(F, length(graphs))) where {F,W,C}

Construct a product graph from multiple input graphs, where each graph can be weighted by a constant. For graphs that are repeated more than once, it adds a power operator to the subgraph to represent the repetition. Moreover, it optimizes any trivial unary operators in the resulting product graph.

Arguments:

• graphs::Vector{Graph{F,W}}: A vector of input graphs to be multiplied.
• constants::AbstractVector: A vector of scalar multiples. If not provided, it defaults to a vector of ones of the same length as graphs.

Returns:

• A new product graph with the unique subgraphs (or powered versions thereof) and the associated constants as subgraph factors.

Example:

Given graphs `g1`, `g2`, `g1` and constants `c1`, `c2`, `c3`, the function computes `(c1*c3)*(g1)^2 * c2*g2`.

function multi_product(g1::Graph{F,W}, g2::Graph{F,W}, c1=F(1), c2=F(1)) where {F,W,C}

Returns a graph representing the multi product c1*g1 * c2*g2. If g1 == g2, it will return a graph representing c1*c2 * (g1)^2 with Power(2) operator.

Arguments:

• g1: first computational graph
• g2: second computational graph
• c1: first scalar multiple (defaults to 1).
• c2: second scalar multiple (defaults to 1).

function name(g::AbstractGraph)

Returns the name of computational graph `g`.

function onechild(g::AbstractGraph)

Returns whether the graph g has only one child (subgraph).

Arguments:

• g::AbstractGraph: graph to be analyzed

open_parenthesis!(graph::G; map::Dict{Int,G}=Dict{Int,G}()) where {G<:AbstractGraph}

Recursively open parenthesis of subgraphs within the given graph `g`with in place form.  The graph eventually becomes 
a single Sum root node with multiple subgraphs that represents multi-product of nodes (not flattened).

Arguments:

• g::AbstractGraph: graph to be modified
• map::Dict{Int,G}=Dict{Int,G}(): A dictionary that maps the id of an original node with its corresponding new node after transformation.

In recursive transform, nodes can be visited several times by different parents. This map keeps track of those visited, and reuse those transformed sub-branches instead of recreating them. parents

function operator(g::AbstractGraph)

Returns the operation associated with computational graph node `g`.

function optimize!(graphs::Union{Tuple,AbstractVector{<:AbstractGraph}}; level=0, verbose=0, normalize=nothing)

In-place optimization of given `graphs`. Removes duplicated leaves, flattens chains, 
merges linear combinations, and removes zero-valued subgraphs. When `level > 0`, also removes duplicated intermediate nodes.

Arguments:

• graphs: A tuple or vector of graphs.
• level: Optimization level (default: 0). A value greater than 0 triggers more extensive but slower optimization processes, such as removing duplicated intermediate nodes.
• verbose: Level of verbosity (default: 0).
• normalize: Optional function to normalize the graphs (default: nothing).

Returns

• Returns the optimized graphs. If the input graphs is empty, it returns nothing.

function optimize(graphs::Union{Tuple,AbstractVector{<:AbstractGraph}}; level=0, verbose=0, normalize=nothing)

Optimizes a copy of given `graphs`. Removes duplicated nodes (when `level > 0`) or leaves, flattens chains, 
merges linear combinations, and removing zero-valued subgraphs.

Arguments:

• graphs: A tuple or vector of graphs.
• level: Optimization level (default: 0). A value greater than 0 triggers more extensive but slower optimization processes, such as removing duplicated nodes.
• verbose: Level of verbosity (default: 0).
• normalize: Optional function to normalize the graphs (default: nothing).

Returns:

• A tuple/vector of optimized graphs.

function orders(g::AbstractGraph)

Returns the derivative orders (::Vector{Int}) of computational graph `g`.

function plot_tree(graph::AbstractGraph; verbose = 0, maxdepth = 6)

Visualize the computational graph as a tree using ete3 python package

#Arguments

• graph::AbstractGraph : the computational graph struct to visualize
• verbose=0 : the amount of information to show
• maxdepth=6 : deepest level of the computational graph to show

function propagator(ops::Union{OperatorProduct,Vector{QuantumOperator}};
    name="", factor=one(_dtype.factor), weight=zero(_dtype.weight), operator=Sum())

Create a Propagator-type FeynmanGraph from given OperatorProduct or Vector{QuantumOperator} ops, including two quantum operators.

function properties(g::AbstractGraph)

Returns additional properties, if any, of the computational graph `g`.

function relabel!(g::FeynmanGraph, map::Dict{Int,Int})

Relabels the quantum operators in `g` and its subgraphs according to `map`.
For example, `map = {1=>2, 3=>2}`` will find all quantum operators with labels 1 and 3, and then map them to 2.

Arguments:

• g::FeynmanGraph: graph to be modified
• map: mapping from old labels to the new ones

function relabel(g::FeynmanGraph, map::Dict{Int,Int})

Returns a copy of `g` with quantum operators in `g` and its subgraphs relabeled according to `map`.
For example, `map = {1=>2, 3=>2}` will find all quantum operators with labels 1 and 3, and then map them to 2.

Arguments:

• g::FeynmanGraph: graph to be modified
• map: mapping from old labels to the new ones

function remove_all_zero_valued_subgraphs!(g::AbstractGraph; verbose=0)

Recursively removes all zero-valued subgraph(s) in-place in the given graph `g`.

Arguments:

• g: An AbstractGraph.
• verbose: Level of verbosity (default: 0).

Returns:

• Optimized graph.

function remove_all_zero_valued_subgraphs!(graphs::Union{Tuple,AbstractVector{<:AbstractGraph}}; verbose=0)

Recursively removes all zero-valued subgraph(s) in-place in the given graphs.

Arguments:

• graphs: A collection of graphs to be processed.
• verbose: Level of verbosity (default: 0).

Returns:

• Optimized graphs.

function remove_duplicated_leaves!(graphs::Union{Tuple,AbstractVector{<:AbstractGraph}}; verbose=0, normalize=nothing, kwargs...)

Removes duplicated leaf nodes in-place from a collection of graphs. It also provides optional normalization for these leaves.

Arguments:

• graphs: A collection of graphs to be processed.
• verbose: Level of verbosity (default: 0).
• normalize: Optional function to normalize the graphs (default: nothing).

function remove_zero_valued_subgraphs!(g::AbstractGraph)

Removes zero-valued (zero subgraph_factor) subgraph(s) of a computational graph `g`. If all subgraphs are zero-valued, the first one (`eldest(g)`) will be retained.

Arguments:

• g::AbstractGraph: graph to be modified

function remove_zero_valued_subgraphs(g::AbstractGraph)

Returns a copy of graph `g` with zero-valued (zero subgraph_factor) subgraph(s) removed.
If all subgraphs are zero-valued, the first one (`eldest(g)`) will be retained.

Arguments:

• g::AbstractGraph: graph to be modified

function replace_subgraph!(g::AbstractGraph, w::AbstractGraph, m::AbstractGraph)

Modifies `g` by replacing the subgraph `w` with a new graph `m`.
For Feynman diagrams, subgraphs `w` and `m` should have the same diagram type, orders, and external indices.

Arguments:

• g::AbstractGraph: graph to be modified
• w::AbstractGraph: subgraph to replace
• m::AbstractGraph: new subgraph

function replace_subgraph(g::AbstractGraph, w::AbstractGraph, m::AbstractGraph)

Creates a modified copy of `g` by replacing the subgraph `w` with a new graph `m`.
For Feynman diagrams, subgraphs `w` and `m` should have the same diagram type, orders, and external indices.

Arguments:

• g::AbstractGraph: graph to be modified
• w::AbstractGraph: subgraph to replace
• m::AbstractGraph: new subgraph

function set_id!(g::AbstractGraph, id)

Update the id of graph `g` to `id`.

function set_name!(g::AbstractGraph, name::AbstractString)

Update the name of graph `g` to `name`.

function set_operator!(g::AbstractGraph, operator::AbstractOperator)

Update the operator of graph `g` to `typeof(operator)`.

function set_operator!(g::AbstractGraph, operator::Type{<:AbstractOperator})

Update the operator of graph `g` to `operator`.

function set_orders!(g::AbstractGraph, orders::AbstractVector)

Update the orders of graph `g` to `orders`.

function set_properties!(g::AbstractGraph, properties)

Update the properties of graph `g` to `properties`.

function set_subgraph!(g::AbstractGraph, subgraph::AbstractGraph, i=1)

Update the `i`th subgraph of graph `g` to `subgraph`.
Defaults to the first subgraph factor if an index `i` is not supplied.

function setsubgraphfactor!(g::AbstractGraph, subgraph_factor, i=1)

Update the `i`th subgraph factor of graph `g` to `subgraph_factor`.
Defaults to the first subgraph factor if an index `i` is not supplied.

function setsubgraphfactors!(g::AbstractGraph, subgraph_factors::AbstractVector, indices::AbstractVector{Int})

Update the specified subgraph factors of graph `g` at indices `indices` to `subgraph_factors`.
By default, calls `set_subgraph_factor!(g, subgraph_factors[i], i)` for each `i` in `indices`.

function setsubgraphfactors!(g::AbstractGraph, subgraph_factors::AbstractVector)

Update the subgraph factors of graph `g` to `subgraphs`.

function set_subgraphs!(g::AbstractGraph, subgraphs::AbstractVector{<:AbstractGraph}, indices::AbstractVector{Int})

Update the specified subgraphs of graph `g` at indices `indices` to `subgraphs`.
By default, calls `set_subgraph!(g, subgraphs[i], i)` for each `i` in `indices`.

function set_subgraphs!(g::AbstractGraph, subgraphs::AbstractVector{<:AbstractGraph})

Update the full list of subgraphs of graph `g` to `subgraphs`.

function set_weight!(g::AbstractGraph, weight)

Update the weight of graph `g` to `weight`.

function standardize_labels!(g::FeynmanGraph)

Finds all labels involved in `g` and its subgraphs and 
modifies `g` by relabeling in standardized order, e.g.,
(1, 4, 5, 7, ...) ↦ (1, 2, 3, 4, ....)

Arguments:

• g::FeynmanGraph: graph to be relabeled

function standardize_labels!(g::FeynmanGraph)

Finds all labels involved in `g` and its subgraphs and returns 
a copy of `g` relabeled in a standardized order, e.g.,
(1, 4, 5, 7, ...) ↦ (1, 2, 3, 4, ....)

Arguments:

• g::FeynmanGraph: graph to be relabeled

function subgraph(g::AbstractGraph, i=1)

Returns a copy of the `i`th subgraph of computational graph `g`.
Defaults to the first subgraph if an index `i` is not supplied.

function subgraph_factor(g::AbstractGraph, i=1)

Returns a copy of the `i`th subgraph factor of computational graph `g`.
Defaults to the first subgraph factor if an index `i` is not supplied.

function subgraph_factors(g::AbstractGraph, indices::AbstractVector{Int})

Returns the subgraph factors of computational graph `g` at indices `indices`.
By default, calls `subgraph_factor(g, i)` for each `i` in `indices`.

function subgraph_factors(g::AbstractGraph)

Returns the subgraph factors of computational graph `g`.

function subgraphs(g::AbstractGraph, indices::AbstractVector{Int})

Returns the subgraphs of computational graph `g` at indices `indices`.
By default, calls `subgraph(g, i)` for each `i` in `indices`.

function subgraphs(g::AbstractGraph)

Returns the subgraphs of computational graph `g`.

function topology(g::FeynmanGraph)

Returns the topology (::Vector{Vector{Int}}) of FeynmanGraph g.

function unary_istrivial(g::AbstractGraph)

Returns true if the unary form of the graph operation of node `g` is trivial.
Otherwise, returns false.

function unique_nodes!(graphs::AbstractVector{<:AbstractGraph})

Identifies and retrieves unique nodes from a set of graphs.

Arguments:

• graphs: A collection of graphs to be processed.

Returns:

• A mapping dictionary from the id of each leaf to the unique leaf node.

function vertex(g::FeynmanGraph, i=1)

Returns the ith vertex (::OperatorProduct) of FeynmanGraph g. Defaults to the first vertex if an index i is not supplied.

function vertices(g::FeynmanGraph)

Returns all vertices (::Vector{OperatorProduct}) of FeynmanGraph g.

function weight(g::AbstractGraph)

Returns the weight of the computational graph `g`.