GAP (AGT) - Chapter 2: Regular graphs

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2 Regular graphs
2.1 Regular graphs
2.1-1 RGParameters
2.1-2 IsRG
2.1-3 IsFeasibleRGParameters
2.2 Edge-regular graphs
2.2-1 ERGParameters
2.2-2 IsERG
2.2-3 IsFeasibleERGParameters
2.3 Strongly regular graphs
2.3-1 SRGParameters
2.3-2 IsSRG
2.3-3 IsFeasibleSRGParameters

### 2 Regular graphs

In this chapter we give functions used to identify graphs with various regularity properties and determine their parameters.

#### 2.1 Regular graphs

A graph \(\Gamma\) is regular with parameters \((v,k)\) if \(\Gamma\) is simple and undirected, it has order \(v\), and every vertex of \(\Gamma\) has degree \(k\).

##### 2.1-1 RGParameters

`‣ RGParameters`( gamma ) | ( function )

Returns: A list or `fail`.

Given a graph gamma, this function returns the regular graph parameters of gamma. If gamma is not a regular graph, the function returns `fail`.

    gap> gamma:=EdgeOrbitsGraph(Group((2,3,4,5)),[[1,2],[2,1]]);;
    gap> RGParameters(gamma);
    fail
    gap> gamma:=HammingGraph(3,4);;
    gap> RGParameters(gamma);
    [ 64, 9 ]

##### 2.1-2 IsRG

`‣ IsRG`( gamma ) | ( function )

Returns: `true` or `false`.

Given a graph gamma, this function returns `true` if gamma is a regular graph, and `false` otherwise.

    gap> gamma:=NullGraph(Group(()),5);;
    gap> IsRG(gamma);
    true
    gap> gamma:=EdgeOrbitsGraph(Group((2,3,4,5)),[[1,2],[2,1]]);;
    gap> IsRG(gamma);
    false
    gap> gamma:=TriangularGraph(6);;
    gap> IsRG(gamma);
    true

##### 2.1-3 IsFeasibleRGParameters

`‣ IsFeasibleRGParameters`( [v, k] ) | ( function )

Returns: `true` or `false`.

Given a list of integers of length 2, [v,k], this function returns `true` if \(( \textit{v}, \textit{k} )\) is a feasible parameter tuple for a regular graph. Otherwise, the function returns `false`.

The tuple \((v, k)\) is a feasible parameter tuple for a regular graph if it satisfies the following well-known conditions:

- \(v>k\geq 0\);

- \(2\) divides \(vk\).

Any regular graph must have parameters that satisfy these conditions (see [BCN89]).

    gap> IsFeasibleRGParameters([15,9]);
    false
    gap> IsFeasibleRGParameters([16,9]);
    true

#### 2.2 Edge-regular graphs

A graph \(\Gamma\) is edge-regular with parameters \((v,k,a)\) if it is regular with parameters \((v,k)\), it has at least one edge, and every pair of adjacent vertices have exactly \(a\) common neighbours.

##### 2.2-1 ERGParameters

`‣ ERGParameters`( gamma ) | ( function )

Returns: A list or `fail`.

Given a graph gamma, this function returns the edge-regular graph parameters of gamma. If gamma is not an edge-regular graph, the function returns `fail`.

    gap> gamma:=NullGraph(Group(()),5);;
    gap> ERGParameters(gamma);
    fail
    gap> gamma:=JohnsonGraph(7,3);;
    gap> ERGParameters(gamma);
    [ 35, 12, 5 ]

##### 2.2-2 IsERG

`‣ IsERG`( gamma ) | ( function )

Returns: `true` or `false`.

Given a graph gamma, this function returns `true` if gamma is an edge-regular graph, and `false` otherwise.

    gap> gamma:=NullGraph(Group(()),5);;
    gap> IsERG(gamma);
    false
    gap> gamma:=JohnsonGraph(7,3);;
    gap> IsERG(gamma);
    true

##### 2.2-3 IsFeasibleERGParameters

`‣ IsFeasibleERGParameters`( [v, k, a] ) | ( function )

Returns: `true` or `false`.

Given a list of integers of length 3, [v,k,a], this function returns `true` if \(( \textit{v, k, a} )\) is a feasible parameter tuple for an edge-regular graph. Otherwise, the function returns `false`.

The tuple \(( v, k, a )\) is a feasible parameter tuple for an edge-regular graph if it satisfies the following well-known conditions:

- \((v,k)\) is a feasible regular graph parameter tuple;

- \(k>a\geq 0\);

- \(2\) divides \(ka\) and \(6\) divides \(vka\);

- \(v-2k+a \geq 0\).

Any edge-regular graph must have parameters which satisfy these conditions (see [BCN89]).

    gap> IsFeasibleERGParameters([15,9,6]);
    false
    gap> IsFeasibleERGParameters([16,9,4]);
    true

#### 2.3 Strongly regular graphs

A graph \(\Gamma\) is strongly regular with parameters \((v,k,a,b)\) if it is edge-regular with parameters \((v,k,a)\), it has at least one pair of distinct non-adjacent vertices, and every pair of distinct non-adjacent vertices have exactly \(b\) common neighbours.

##### 2.3-1 SRGParameters

`‣ SRGParameters`( gamma ) | ( function )

Returns: A list or `fail`.

Given a graph gamma, this function returns the strongly regular graph parameters of gamma. If gamma is not a strongly regular graph, the function returns `fail`.

    gap> gamma:=CompleteGraph(Group(()),5);;
    gap> SRGParameters(gamma);
    fail
    gap> gamma:=JohnsonGraph(5,3);;
    gap> SRGParameters(gamma);
    [ 10, 6, 3, 4 ]

##### 2.3-2 IsSRG

`‣ IsSRG`( gamma ) | ( function )

Returns: `true` or `false`.

Given a graph gamma, this function returns `true` if gamma is a strongly regular graph, and `false` otherwise.

    gap> gamma:=CompleteGraph(Group(()),5);;
    gap> IsSRG(gamma);
    false
    gap> gamma:=JohnsonGraph(5,3);;
    gap> IsSRG(gamma);
    true

##### 2.3-3 IsFeasibleSRGParameters

`‣ IsFeasibleSRGParameters`( [v, k, a, b] ) | ( function )

Returns: `true` or `false`.

Given a list of integers of length 4, [v,k,a,b], this function returns `true` if \(( \textit{v, k, a, b} )\) is a feasible parameter tuple for a strongly regular graph. Otherwise, this function returns `false`.

The tuple \((v,k,a,b)\) is a feasible parameter tuple for a strongly regular graph if it satisfies the following well-known conditions:

- \((v,k,a)\) is a feasible edge-regular graph parameter tuple;

- \(k\geq b\);

- \((v-k-1)b = k(k-a-1)\);

- \(v-2-2k+b \geq 0\);

- the formulae for the multiplicities of the eigenvalues of a strongly regular graph with these parameters evaluate to positive integers (see [BH11]).

Any strongly regular graph must have parameters which satisfy these conditions (see [BCN89]).

    gap> IsFeasibleSRGParameters([15,9,4,7]);
    false
    gap> IsFeasibleSRGParameters([10,3,0,1]);
    true

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