On algebraic characterization of SSC of the Jahangir’s graph 𝓙_n,m

Definition 2.1

A spanning tree of a simple connected finite graphG(V, E) is a subtree ofGthat contains every vertex ofG.

We represent the collection of all edge-sets of the spanning trees ofGbys(G), in other words;

Lemma 2.2

Let G = (V, E) be a simple finite connected graph containingmcycles. Then its spanning tree contains exactly |E| − medges.

Proof

A spanning tree of a graph is its spanning subgraph containing no cycles and no disconnection. If G is a unicyclic graph then deletion of one edge from it results in a spanning tree. If more than one edge is removed from the cycle in G then a disconnection is obtained which is not a spanning tree. Therefore, spanning tree has exactly |E| − 1 edges.

If G has m disjoint cycles in it i.e. cycles sharing no common edges, then its spanning tree is obtained by removing exactly m edges from it, one from each of its cycle. Therefore, its spanning tree has |E| − m edges in it.

If any two cycles of G share one or more common edges and remaining are disjoint cycles, then one edge is needed to be removed from each cycle of G to obtain a spanning tree. However, if a common edge between two cycles is removed then exactly one edge from non common edges must be removed of the resulting big cycle. Therefore, its spanning tree has |E| − m edges in it. This can be extended to any number of cycles in G sharing common edges. This completes the proof. □

Definition 2.3

A simplicial complexΔover a finite set [n] = {1, 2, …, n} is a collection of subsets of [n], with the property that {i} ∈ Δfor all i ∈ [n], and if F ∈ ΔthenΔwill contain all the subsets ofF (including the empty set). An element ofΔis called a face ofΔ, and the dimension of a faceFofΔis defined as |F| − 1, where |F| is the number of vertices ofF. The maximal faces ofΔunder inclusion are called facets ofΔ. The dimension of the simplicial complexΔis:

We denote the simplicial complexΔwith facets {F₁, …, F_q} by

Definition 2.4

For a simplicial complexΔhaving dimensiond, its f − vector is ad + 1-tuple, defined as:

wheref_idenotes the number ofi − dimensional faces ofΔ.

Definition 2.5 (Spanning Simplicial Complex)

LetG(V, E) be a simple finite connected graph ands(G) = {E₁, E₂, …, E_t} be the edge-sets of all possible spanning trees ofG(V, E), then we defined (in [1])a simplicial complexΔ_s(G) onEsuch that the facets ofΔ_s(G) are precisely the elements ofs(G), we callΔ_s(G) as thespanning simplicial complex ofG(V, E). In other words;

Definition 3.1

LetC_i1, C_i2, …, C_ikbe consecutive cycles in the Jahangir’s graph 𝓙_2,m. Then the cycle obtained by deleting the common edges between the consecutive cyclesC_i1, C_i2, …, C_ikis a new cycle of the Jahangri’s graph 𝓙_2,mis denoted byC_{i1,i2, …, ik}. The cardinality count of the number of edges in the cycleC_i1,i2,…,ikis denoted byβ_i1,i2,…,ik = |C_i1,i2,…,ik|.

Lemma 3.2 (Characterization of 𝓙_2,m)

Let 𝓙_2,mbe the graph with the edgesEas is defined in eq. (1) andC₁, C₂, ⋯, C_mbe itsmconsecutive cycles of equal lengths, then the total number of cycles in the graph are

such thatβ_i1,i2,…,ik = 2(k + 1).

Proof

The Jahangir’s graph 𝓙_2,m contains more than just m consecutive cycles. The remaining cycles can be obtained by deleting the common edges between any number (included) of consecutive cycles and getting a cycle by their remaining edges. The cycle obtained in this way by adjoining consecutive cycles C_i1, C_i2, …, C_ik is denoted by C_{i1,i2, …, ik}. Therefore, we get the following cycles

Combining these with m cycles given we have total cycles in the graph 𝓙_2,m,

such that i_j+1 = i_j + 1 if i_j ≠ m and i_j+1 = 1 if i_j = m.

Now for a fixed value of k, simple counting reveals that the total number of cycles C_i1,i2,…,ik is m for i_k <m. Hence the total number of cycles in 𝓙_2,m is τ. Also it is clear from the construction above that C_i1,i2,…,ik is obtained by deleting common edges between consecutive cycles C_i1,C_i2,…,C_ik which are k − 1 in number. Therefore, the order of the cycle C_i1,i2,…,ik is obtained by adding orders of all C_i1,C_i2, …,C_ik that is, 4k and subtracting 2(k − 1) from it, since the common edges are being counted twice in sum. This implies

 □

Proposition 3.3

Let 𝓙_2,mbe the graph with the edges E as defined ineq. (1) such that {u₁, u₂, …,u_p} ⊆ {v₁,v₂, …,v_q} then we have

Proof

Since the cycles C_u1,u2,…,up and C_v1,v2,…,vq are obtained by deleting the common edges between cycles C_u1,C_u2, …,C_up and C_v1,C_v2, …,C_vq respectively, therefore, {u₁, u_p} ⊈ {v₁,v_q} implies {u₁,u₂, …,u_p} ⊂ {v₁, v₂, …, v_q}. Hence, the intersection C_u1,u2,…,up ⋂ C_v1,v2,…,vq will contain only the non common edges of the cycle C_u1,u2,…,up excluding its two edges common with the cycles on its each end. This gives the order of intersection in this case β_u1,u2,…,up − 2. The remaining cases can be visualized in a similar manner. □

Proposition 3.4

Let 𝓙_2,mbe the graph with the edges E as defined ineq. (1) such that {u₁, u₂, …, u_σ} ⊆ {v₁, v₂, …, v_q} andu_t ∈ {u₁, u₂, …, u_p} & u_t−1 →u_twitht ≤ σ < pthen we have

Proof

Here, the cycles C_u1, C_u2, …, C_uσ are amongst σ consecutive adjoining cycles of the cycle C_u1,u2,…,up which are also overlapping with the σ consecutive adjoining cycles of the cycle C_v1,v2,…,vq. If the adjoining cycle C_u1 of the cycle C_u1,u2,…,up overlaps with the first adjoining cycle C_v1 of the cycle C_v1,v2,…,vq and the adjoining cycles C_vq and C_u1 are consecutive then by previous proposition the order of the intersection C_u1,u2,…,up ⋂ C_v1,v2,…,vq is indeed β_{u1,u2, …,uσ} − 1. Similarly, if the adjoining cycles C_vq and C_u1 are not consecutive then they will have no common edge and the use of proposition 3.3 gives the order of the intersection C_u1,u2,…,up ⋂ C_v1,v2,…,vq as β_{u1,u2, …,uσ} − 2. Similar can be done for the remaining cases. □

Remark 3.5

The case when there exists at₀ <σ < psuch thatu_t0−1 ↛ u_t0in above proposition i.e., when cyclesC_u1, C_u2, …, C_ut0−1, C_ut0, …, C_uσare not amongstσconsecutive adjoining cycles of the cycleC_u1,u2,…,up, the order of the intersectionC_u1,u2,…,up ⋂ C_v1,v2,…,vqcan be calculated by applyingproposition 3.4on the overlapping portions.

Proposition 3.6

Let 𝓙_2,mbe the graph with the edgesEas defined in (1) such that {u₁,u₂, …,u_p} ⋂ {v₁,v₂, …,v_q} = φandp ≤ q. Then we have

Proof

In this case the adjoining cycles of C_u1,u2,…,up and C_v1,v2,…,vq have no common cycle. However, if the adjoining cycle on one of the extreme ends of the cycle C_u1,u2,…,up is consecutive with the adjoining cycles on one of the extreme ends of the other cycle C_v1,v2,…,vq then the intersection C_u1,u2,…,up ⋂ C_v1,v2,…,vq will have only one edge. The remaining cases are easy to see. □

Proposition 3.7

A subsetE(T_{(j1i1,j2i2,…,jmim)}ofEwithj_αi_α ≠ j_α 1 for allαwill belong tos(𝓙_2,m) if and only if

Proof

𝓙_2,m is a graph with cycles C₁, C₂, …, C_m and e₁₁, e₂₁, …, e_m1 are the common edges between the consecutive cycles. The cutting down process explains we need to remove exactly m edges, keeping the graph connected and no cycles and no isolated edge left and no isolated vertices left in the graph. Therefore, in order to obtain a spanning tree of 𝓙_2,m with none of common edges e₁₁, e₂₁, …, e_m1 to be removed, we need to remove exactly one edge from the non common edges from each cycle. This explains the proof of the proposition. □

Proposition 3.8

A subsetE(T_{(j1i1,j2i2,…,jmim)}ofEwithj_αi_α = j_α 1 for anyαwill belong tos(𝓙_2,m) if and only if

where, {e_j1i1, e_j2i2, …, e_jmim} will contain exactly one edge fromC_{(jα−1)(jα)} ∖ {e_(jα−1)1, e_(jα+1)1} other thane_jα1.

Proof

For a spanning tree of 𝓙_2,m such that exactly one common edge e_jα1 is removed, we need to remove precisely m-1 edges from the remaining edges using the cutting down process. However, we cannot remove more than one edge from the non common edges of the cycle C_{(jα−1)(jα)} (since this will result a disconnected graph. This explains the proof of the above case.

Proposition 3.9

A subsetE(T_{(j1i1,j2i2,…,jmim)}) ⊂ E, wherej_αi_α = j_α 1 forα ∈ {r₁,r₂, …, r_ρ} ⊂ {1, 2, …, m}, will belong tos(𝓙_2,m) if and only if it satisfies any of the following:

1. ife_jr11, e_jr21, …, e_jrρ1are common edges from consecutive cycles then

   E(T(j1i1,j2i2,…,jmim))=E∖{ej1i1,ej2i2,…,ejmim}

   such that {e_j1i1, e_j2i2, …, e_jmim} will contain exactly exactly one edge fromC_{jr0jr1…jrρ}other thane_jr11, e_j(r2)1, …, e_jrρ1, wherej_r0 → j_r1.

2. if none ofe_jr11, e_jr21, …, e_jrρ1are common edges from consecutive cycles then

   E(T(j1i1,j2i2,…,jmim))=E∖{ej1i1,ej2i2,…,ejmim}

   such that for each edgee_jrt1proposition 3.6holds.

3. if some ofe_jr11,e_jr21, …, e_jrρ1are common edges from consecutive cycles then

   E(T(j1i1,j2i2,…,jmim))=E∖{ej1i1,ej2i2,…,ejmim}

   such thatproposition 3.9.1is satisfied for the common edges of consecutive cycles andproposition 3.9.2is satisfied for remaining common edges.

Proof

For the case 1, we need to obtain a spanning tree of 𝓙_2,m such that |r_ρ − r₁|_m common edges must be removed from ρ consecutive cycles C_jr1, C_jr2, …, C_jrρ. The remaining m − |r_ρ − r₁|_m edges must be removed in such a way that exactly one edge is removed from the non common edges of the adjoining cycles C_jr0, C_jr1, …, C_jrρ and the remaining m − |r_ρ − r₁|_m cycles of the graph 𝓙_2,m. This concludes the case.

The remaining cases of the proposition can be visualised in a similar manner using the propositions 3.7 and 3.8. This completes the proof. □

Remark 3.10

If we denote the disjoint classes of subsets ofEdiscussed inpropositions 3.7,3.8and3.9by 𝓒_𝓙1, 𝓒_𝓙2, 𝓒_𝓙3a, 𝓒_𝓙3b, 𝓒_𝓙3crespectively, then, we can writes(𝓙_2,m) as follows:

Proposition 3.11

LetΔ_s(𝓙_2,m) be a spanning simplicial complex of the graph 𝓙_2,m, then thedim(Δ_s(𝓙_2,m)) = 2m − 1 withf − vectorf(Δ_s(𝓙_2,m)) = (f₀, f₁, ⋯, f_2m−1) and

where 0 ≤ i ≤ 2m − 1 I = {i₁i₂… i_k|i_j ∈ {1, 2, …, m} and 1 ≤ k ≤ m such that i_j+1 = i_j + 1 ifi_j ≠ m and i_j+1 = 1 ifi_j = m} andCIt = {Subsets of I of cardinality t}.

Proof

Let E be the edge set of 𝓙_2,m and 𝓒_𝓙1, 𝓒_𝓙2, 𝓒_𝓙3a, 𝓒_𝓙3b, 𝓒_𝓙3c are disjoint classes of spanning trees of 𝓙_2,m then from propositions 3.7, 3.8, 3.9 and the remark 3.10 we have

Therefore, by definition 2.5 we can write Δ_s (𝓙_2,m) = 〈𝓒_𝓙1 ⋃𝓒_𝓙2 ⋃𝓒_𝓙3a ⋃𝓒_𝓙3b ⋃𝓒_𝓙3c〉. Since each facet Ê_{(j1i1,j2i2,…,jmim)} = E(T_{(j1i1,j2i2,…,jmim}) is obtained by deleting exactly m edges from the edge set of 𝓙_2,m, keeping in view the propositions 3.7, 3.8 and 3.9, therefore dimension of each facet is the same i.e., 2m − 1 (since |Ê_{(j1i1,j2i2,…,jmim)}| = 2m) and hence dimension of Δ_s(𝓙_2,m) will be 2m − 1.

Also it is clear from the definition of Δ_s(𝓙_2,m) that it contains all those subsets of E which do not contain the given sets of cycles {e_k1, e_k2, e_k3, e_(k+1)1} for k ∈ {1, 2, …, m − 1} and {e_m1, e_m2, e_m3, e₁₁} in graph as well as any other cycle in the graph 𝓙_2,m.

Now by Lemma 3.2 the total cycles in the graph 𝓙_2,m are

such that i_j+1 = i_j + 1 if i_j ≠ m and i_j+1 = 1 if i_j = m, and their total number is τ. Let F be any subset of E of order i+1 such that it does not contain any C_i1,i2,…,iki_j ∈ {1, 2, …, m} and 1 ≤ k ≤ m, in it. The total number of such F is indeed f_i for 0 ≤ i ≤ 2m − 1. We use inclusion exclusion principle to find this number. Therefore, f_i = Total number of subsets of E of order i + 1 not containing C_i1,i2,…,iki_j ∈ {1, 2, …, m} and 1 ≤ k ≤ m such that i_j+1 = i_j + 1 if i_j ≠ m and i_j+1 = 1 if i_j = m.

Therefore, using these notations and applying Inclusion Exclusion Principle we can write, f_i = (Total number of subsets of E of order i + 1) − ∑{i1}∈CI1 (subset of E of order i + 1 containing C_is for s = 1) + ∑{i1,i2}∈CI2 (subset of E of order i + 1 containing both C_is for all 1 ≤ s ≤ 2 ) − ⋯ + (−1)^τ∑{i1,i2,…,iτ}∈CIτ (subset of E of order i + 1 simultaneously containing each C_is for all 1 ≤ s ≤ τ).

This implies

This implies

Example 3.12

LetΔ_s (𝓙_2,3) be a spanning simplicial complex of the Jahangir’s graph 𝓙_2,mgiven inFigure 1, then thedim(Δ_s (𝓙_2,3)) = 5 andτ = 3² = 9. Therefore, f−vectorsf(Δ_s (𝓙_2,3)) = (f₀, f₁, …, f₅) and

where 0 ≤ i ≤ 5.

Theorem 3.13

LetΔ_s(𝓙_2,m) be the spanning simplicial complex of 𝓙_2,m, then the Hilbert series of the Face ringk[Δ_s(𝓙_2,m)] is given by,

Proof

From [14], we know that if Δ is a simplicial complex of dimension d and f(Δ) = (f₀, f₁, …,f_d) its f-vector, then the Hilbert series of the face ring k[Δ] is given by

By substituting the values of f_i’s from Proposition 3.11 in this above expression, we get the desired result.

Definition 4.1

([2]). Let I ⊂ S = k[x₁,x₂, …,x_n] be a monomial ideal. We say thatIhas linear residuals, if there exists an ordered minimal monomial system of generators {m₁, m₂, …,m_r} ofIsuch that Res(I_i) is minimally generated by linear monomials for all 1 < i ≤ r, where Res(I_i) = {u₁,u₂, …, u_i−1} such thatu_k=migcd(mk,mi)for all 1 ≤ k ≤ i − 1.

Theorem 4.2

([2]). LetΔbe a simplicial complex of dimensiondover [n]. ThenΔwill be a shellable if and only ifI_𝓕(Δ) has linear residuals.

Corollary 4.3

([2]). If the facet idealI_𝓕(Δ) of a pure simplicial complexΔover [n] has linear residuals, then the face ringk[Δ] is Cohen Macaulay.

Theorem 4.4

The face ring ofΔ_s(𝓙_2,m) is Cohen-Macaulay.

Proof

By corollary 4.3, it is sufficient to show that I_𝓕(Δ_s(𝓙_2,m)) has linear residuals in S = k[x₁₁, x₁₂, x₁₃, x₂₁, x₂₂, x₂₃, x₃₁, …,x_m1, x_m2, x_m3]. By propositions 3.7, 3.8, 3.9 and the remark 3.10, we have

Therefore,

and hence we can write,

Here, I_𝓕(Δ_s(𝓙_2,m)) is a pure monomial ideal of degree 2m − 1 with x_{Ê(j1i1,j2i2,…,jmim)} as the product of all variables in S except x_j1i1, x_j2i2, …, x_jmim. Now we will show that I_𝓕(Δ_s(𝓙_2,m)) has linear residuals with respect to the following orders in its monomials:

More explicitly, the monomials {x_{Ê(j1i1,j2i2,…,jmim)}∣ i_r1 ≠ 1; 1 ≤ r₁ ≤ m & i_k = 1; k ≠ r₁} in the order 2, consists of monomials of the form x_{Ê(11,21,…,(m−1)1,jmim)}, x_{Ê(11,21,…,j(m−1)i(m−1),m1)}, …, x_{Ê(11,j2i2,31,…,(m−1)1,m1)}, x_{Ê(j1i1,21,…,(m−1)1,m1)}, where i_k ∈ {2,3} and 1 ≤ j_k ≤ m. Similarly, other monomials in order 2. Let us put

For instance, for r₁ = m in Res(x_{Ê(j1i1,j2i2,…,jmmim)}) we have, Res(x_{Ê(11,21,…,(m−1)1,jmim)}) ={xE^(11,21,…,(m−1)1,jmim)gcd(mk,xE^(11,21,…,(m−1)1,jmim))} where, m_k in this case are all the monomials in S of the form x_{Ê(j1i1,j2i2,…,jmim)} where i_m ≠ m and j_m = 2, 3. Since all the monomials m_k differ from x_{Ê(11,21,…,(m−1)1,jmim)} at only one position, therefore, Res(x_{Ê(11,21,…,(m−1)1,jmim)}) have all linear terms i.e., Res(x_{Ê(11,21,…,(m−1)1,jmim)}) is minimally generated by linear monomials.

Continuing the same process the order 2 of the monomials of I_𝓕(Δ_s(𝓙_2,m)) guarantees that Res(x_{Ê(j1i1,j2i2,…,jmim)}) is minimally generated by linear monomials for all x_{Ê(j1i1,j2i2,…,jmim)} ∈ I_𝓕(Δ_s(𝓙_2,m)). Hence, I_𝓕(Δ_s(𝓙_2,m)) has linear residuals, and by Corollary 4.3Δ_s(𝓙_2,m) is Cohen-Macaulay. □