A Study on Sprout Graphs

Sprout graphs are finite directed graphs matured over a finite subset of the non-negative time line. A simple undirected connected graph on at least two vertices is required to construct an infant graph to mature from. The maxi-max arc-weight principle and the maxi-min arc-weight principle are introduced in the paper. These principles are used to determine the maximum and minimum maturity weight of a sprout graph.


Introduction
This is an introduction to the concept of Sprouting Graphs, which are finite directed graphs cloned on a time line from genetically known simple undirected connected graphs on at least two vertices. The idea of sprouting could remind of neurogical growth of good or malicious networks. Consider a simple undirected connected graph G on n ≥ 2 vertices. Let all vertices carry the default label v. Randomly indice the vertices v i , i = 1, 2, 3, ..., n. Call the indice pattern ℓ α . Note that a graph on n vertices has n! possible indice patterns.
Also reduce an edge v i v j ∈ E(G) to a sprout ij if and only if i < j or ji if and only if j < i. The directed Sprouting Graph G s t∈T ℓα evolves on the time line t ∈ T ℓα ⊆ {0, 1, 2, 3, ...., m} with m = max ∀i,j |i − j| v i ,v j ∈V (G) as per definition 1.1. It implies that at t = 0 the graph G has been reduced to an edgeless graph with sprouts ij, ∀v i v j ∈ V (G) and i < j attached to vertex v i . Note that the number of sprouts attached to v i at t = 0 is equal to d + (v i ) in G s t=m .
Definition 1.1. In a Sprouting Graph G s t∈T ℓα an arc, (v i , v j ) evolves from the sprout ij and arcs at time are called adult vertices, and the set of adult vertices is denoted, A(G s t=m ). Lemma 1.1. All matured Sprouting Graphs G s t=m , cloned from a simple undirected connected graph G on at least two vertices have at least one adult vertex.
Proof. Since a sprout ni, i ≤ (n − 1) can never exist, we have d( and if all arcs are labelled e ℓ , ℓ = 1, 2, 3, ..., ǫ(G s t=m ) the maturity weight of G s t=m , is defined to be It follows easily that w(e ℓ ) is not always a constant and depends on the random indicing of the vertices of a simple undirected connected graph G, so min w(e ℓ ).

Sprouting of Complete Graphs, Paths and Cycles
Since complete graphs, paths and possibly cycles amongst others, form part of the skeleton of all graphs, it is thought that the introduction to sprouting results within these classes of graphs will entice further results.
Proposition 2.1. For the complete graph K n we have that mw(K s n,t=m ) = Proof. Consider the complete graph K n . Indice any vertex as v 1 and randomly indice the other vertices v 2 , v 3 , ..., v n . Now consider the matured Sprouting Graph K s n,t=m . Regardless of the random indicing of the vertices we have the following arc-weights, By summing all columns carrying equal arc-weights, across all rows the result mw(K s n,t=m ) = (n − i), for all indice patterns ℓ α , (n! indice patterns).
Proof. Indice any vertex of K n with the number 1. Randomly indice the remaining vertices 2, 3, ..., n. Due to complete connectivity the results follows trivially.
Corollary 2.3. The complete sprouting graph K s n,t=m has a unique adult vertex, v n so A = {v n }.
Proof. Since any vertex v i , i < n is always linked to v n , the vertex v i will always have Since it contradicts definition 1.2 the result follows from Lemma 1.1.
Lemma 2.4. For a simple undirected connected graph G on n vertices we have max(mw(G s t=m )) ≤ min(mw(K s n,t=m )) = max(mw(K s n,t=m )).
Proof. It follows from Corollary 2.2 that min(mw(K s n,t=m )) = max(mw(K s n,t=m )). Since |ǫ(K n )| ≥ |ǫ(G)|, G on n vertices, the removal of edges from K n to obtain G results in reducing the corresponding terms in the summation mw(K s n,t=m ) = (n − i), to zero. Therefore, max(mw(G s t=m )) ≤ min(mw(K s n,t=m )) = max(mw(K s n,t=m )). Proposition 2.5. For the path P n , n ≥ 2, we have that min(mw(P s n,t=m )) = n − 1, and max(mw(P s n,t=m )) = Proof. (Part 1): Conceptualise the path P n , n ≥ 2 as the graph with the n vertices seated on a horisontal line. Indice the leftmost end vertex, v 1 and the rightmost end vertex, v n .
Proof. Follows directly from Proposition 2.5.
Proposition 2.7. For the cycle C n , n ≥ 4, we have that min(mw(C s n,t=m )) = 2(n − 1), and max(mw(C s n,t=m )) = Proof. (Part 1): Conceptualise the path C n , n ≥ 4 as the graph with the n vertices seated on the circumference of a circle with a vertex seated centre at the top. Indice the top vertex, v 1 and indice the other vertices clockwise v 2 , v 3 , ..., v n . Call the indice pattern ℓ 1 . Clearly, the acrc weights, w(v i , v j ) = 1 except for, w(v 1 , v n ) = n − 1 hence T ℓ 1 = {0, 1, (n − 1)}. Therefore all arcs having arc-weight = 1, arc at t = 1. Since there are exactly (n − 1) such arcs in C s n,t=1 and the arc (v 1 , v n ) arcs at t = (n − 1), the first part of the result follows.
Proof. Follows directly from Proposition 2.7.
It has been established that if two different random indice patterns of a graph G say ℓ 1 and ℓ 2 result in T ℓ 1 and T ℓ 2 respectively, such that T ℓ 1 = T ℓ 2 then, ). However, it is conjectured by Kok that if T ℓ 1 ⊂ T ℓ 2 then mw(G s t=m ℓ 1 ) < mw(G s t=m ℓ 2 ).
(ii) By indicing the nodal vertex, v 1 and the leafs, v 2 , v 3 , ..., v n+1 the result follows similarly to the proof of (i).
[Open problem: Is it possible to always find an indice pattern for the vertices of a simple undirected connected graph, resulting in a Sprouting Graph with a unique adult vertex ?] [Open problem: Prove or disprove Kok's pattern conjecture namely, if two different random indice patterns of a graph G say ℓ 1 and ℓ 2 result in T ℓ 1 and T ℓ 2 respectively, then T ℓ 1 ⊂ T ℓ 2 ⇒ mw(G s t=m ℓ 1 ) < mw(G s t=m ℓ 2 ).] [Open problem: What is min(mw(K s (r 1 ,r 2 ,r 3 ,...,rn),t=m )) and max(mw(K s (r 1 ,r 2 ,r 3 ,...,rn),t=m )) with K (r 1 ,r 2 ,r 3 ,...,rn) , 2 ≤ r 1 ≤ r 2 ≤ r 3 ≤ ... ≤ r n , ∀r i ∈ N, being a complete n-partite graph ?] [Open problem: Describe an algorithm to determine max(mw(T s t=m )), T a tree.] [Open problem: If possible, rework the proof of Theorem 3.1 to improve its mathematical elegancy.] [Open problem: Conceptualise a Sprouting Graph G on n vertices and indiced i j such that, i 1 < i 2 , .... < i n and i j ∈ N. Determine max(mw(G s t=m )) and min(mw(G s t=m )) in general.] Open access: This research proposal is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution and reproduction in any medium, provided the original author(s) and the source are credited.