In this section, we give some background and preliminaries of the topic and define some important notions to make this paper self-contained. However, for more details of the notions we refer the reader to [3, 4, 5, 6, 7, 13, 14].

#### Definition 2.1

*A spanning tree of a simple connected finite graph* *G*(*V*, *E*) *is a subtree of* *G* *that contains every vertex of* *G*.

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

$$\begin{array}{}{\displaystyle s(G):=\{E({T}_{i})\subset E,\phantom{\rule{thinmathspace}{0ex}}where\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{T}_{i}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}is\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}a\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}spanning\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}tree\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}of\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}G\}.}\end{array}$$

#### Lemma 2.2

*Let G* = (*V*, *E*)* be a simple finite connected graph containing* *m* *cycles*. *Then its spanning tree contains exactly* |*E*| − *m* *edges*.

#### 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. □

Applying Lemma 2.2, we can obtain the spanning tree of the Jahangir’s graph 𝓙_{2,m} by removing exactly *m* edges from it keeping in view the following:

–

Not more than one edge can be removed from the non common edges of any cycle.

–

If a common edge between two or more consecutive cycles is removed then exactly one edge must be removed from the resulting big cycle.

–

Not all common edges can be removed simultaneously.

This method is referred as the *cutting*-*down method*. For example, by using the *cutting*-*down method* for the graph 𝓙_{2,3} given in Fig. 1 we obtain:

*s*(𝓙_{2,3}) = {{*e*_{11}, *e*_{21}, *e*_{31}, *e*_{12}, *e*_{22}, *e*_{32}}, {*e*_{11}, *e*_{21}, *e*_{31}, *e*_{12}, *e*_{22}, *e*_{33}}, {*e*_{11}, *e*_{21}, *e*_{31}, *e*_{12}, *e*_{23}, *e*_{32}}, {*e*_{11}, *e*_{21}, *e*_{31}, *e*_{12}, *e*_{23}, *e*_{33}}, {*e*_{11}, *e*_{21}, *e*_{31}, *e*_{13}, *e*_{22}, *e*_{32}}, {*e*_{11}, *e*_{21}, *e*_{31}, *e*_{13}, *e*_{22}, *e*_{33}}, {*e*_{11}, *e*_{21}, *e*_{31}, *e*_{13}, *e*_{23}, *e*_{32}}, {*e*_{11}, *e*_{21}, *e*_{31}, *e*_{13}, *e*_{23}, *e*_{33}}, {*e*_{21}, *e*_{31}, *e*_{32}, *e*_{33}, *e*_{12}, *e*_{22}}, {*e*_{21}, *e*_{31}, *e*_{32}, *e*_{33}, *e*_{12}, *e*_{23}}, {*e*_{21}, *e*_{31}, *e*_{32}, *e*_{33}, *e*_{13}, *e*_{22}}, {*e*_{21}, *e*_{31}, *e*_{32}, *e*_{33}, *e*_{13}, *e*_{23}}, {*e*_{21}, *e*_{31}, *e*_{12}, *e*_{13}, *e*_{33}, *e*_{22}}, {*e*_{21}, *e*_{31}, *e*_{12}, *e*_{13}, *e*_{33}, *e*_{23}}, {*e*_{21}, *e*_{31}, *e*_{12}, *e*_{13}, *e*_{32}, *e*_{22}}, {*e*_{21}, *e*_{31}, *e*_{12}, *e*_{13}, *e*_{32}, *e*_{23}}, {*e*_{11}, *e*_{31}, *e*_{12}, *e*_{13}, *e*_{22}, *e*_{32}}, {*e*_{11}, *e*_{31}, *e*_{12}, *e*_{13}, *e*_{22}, *e*_{33}}, {*e*_{11}, *e*_{31}, *e*_{12}, *e*_{13}, *e*_{23}, *e*_{32}}, {*e*_{11}, *e*_{31}, *e*_{12}, *e*_{13}, *e*_{23}, *e*_{33}}, {*e*_{11}, *e*_{31}, *e*_{22}, *e*_{23}, *e*_{13}, *e*_{32}}, {*e*_{11}, *e*_{31}, *e*_{22}, *e*_{23}, *e*_{13}, *e*_{33}}, {*e*_{11}, *e*_{31}, *e*_{22}, *e*_{23}, *e*_{12}, *e*_{32}}, {*e*_{11}, *e*_{31}, *e*_{22}, *e*_{23}, *e*_{12}, *e*_{33}}, {*e*_{11}, *e*_{21}, *e*_{23}, *e*_{22}, *e*_{32}, *e*_{12}}, {*e*_{11}, *e*_{21}, *e*_{23}, *e*_{22}, *e*_{32}, *e*_{13}}, {*e*_{11}, *e*_{21}, *e*_{23}, *e*_{22}, *e*_{33}, *e*_{12}}, {*e*_{11}, *e*_{21}, *e*_{23}, *e*_{22}, *e*_{33}, *e*_{13}}, {*e*_{11}, *e*_{21}, *e*_{32}, *e*_{33}, *e*_{22}, *e*_{12}}, {*e*_{11}, *e*_{21}, *e*_{32}, *e*_{33}, *e*_{22}, *e*_{13}}, {*e*_{11}, *e*_{21}, *e*_{32}, *e*_{33}, *e*_{23}, *e*_{12}}, {*e*_{11}, *e*_{21}, *e*_{32}, *e*_{33}, *e*_{23}, *e*_{13}}, {*e*_{11}, *e*_{13}, *e*_{22}, *e*_{23}, *e*_{32}, *e*_{33}}, {*e*_{11}, *e*_{12}, *e*_{22}, *e*_{23}, *e*_{32}, *e*_{33}}, {*e*_{11}, *e*_{12}, *e*_{13}, *e*_{23}, *e*_{32}, *e*_{33}}, {*e*_{11}, *e*_{12}, *e*_{13}, *e*_{22}, *e*_{32}, *e*_{33}}, {*e*_{11}, *e*_{12}, *e*_{13}, *e*_{22}, *e*_{23}, *e*_{33}}, {*e*_{11}, *e*_{12}, *e*_{13}, *e*_{22}, *e*_{23}, *e*_{32}}, {*e*_{21}, *e*_{13}, *e*_{22}, *e*_{23}, *e*_{32}, *e*_{33}}, {*e*_{21}, *e*_{12}, *e*_{22}, *e*_{23}, *e*_{32}, *e*_{33}}, {*e*_{21}, *e*_{12}, *e*_{13}, *e*_{23}, *e*_{32}, *e*_{33}}, {*e*_{21}, *e*_{12}, *e*_{13}, *e*_{22}, *e*_{32}, *e*_{33}}, {*e*_{21}, *e*_{12}, *e*_{13}, *e*_{22}, *e*_{23}, *e*_{33}}, {*e*_{21}, *e*_{12}, *e*_{13}, *e*_{22}, *e*_{23}, *e*_{32}}, {*e*_{31}, *e*_{13}, *e*_{22}, *e*_{23}, *e*_{32}, *e*_{33}}, {*e*_{31}, *e*_{12}, *e*_{22}, *e*_{23}, *e*_{32}, *e*_{33}}, {*e*_{31}, *e*_{12}, *e*_{13}, *e*_{23}, *e*_{32}, *e*_{33}}, {*e*_{31}, *e*_{12}, *e*_{13}, *e*_{22}, *e*_{32}, *e*_{33}}, {*e*_{31}, *e*_{12}, *e*_{13}, *e*_{22}, *e*_{23}, *e*_{33}}, {*e*_{31}, *e*_{12}, *e*_{13}, *e*_{22}, *e*_{23}, *e*_{32}}}.

#### Definition 2.3

*A simplicial complex* *Δ* *over a finite set* [*n*] = {1, 2, …, *n*} *is a collection of subsets of* [*n*], *with the property that* {*i*} ∈ *Δ* *for all i* ∈ [*n*], *and if F* ∈ *Δ* *then* *Δ* *will contain all the subsets of* *F (including the empty set)*. *An element of* *Δ* *is called a face of* *Δ*, *and the dimension of a face* *F* *of* *Δ* *is defined as* |*F*| − 1, *where* |*F*| *is the number of vertices of* *F*. *The maximal faces of* *Δ* *under inclusion are called facets of* *Δ*. *The dimension of the simplicial complex* *Δ* *is*:

$$\begin{array}{}{\displaystyle dim\mathit{\Delta}=max\{dimF|F\in \mathit{\Delta}\}.}\end{array}$$

*We denote the simplicial complex* *Δ* *with facets* {*F*_{1}, …, *F*_{q}} *by*

$$\begin{array}{}{\displaystyle \mathit{\Delta}=\u3008{F}_{1},\dots ,{F}_{q}\u3009}\end{array}$$

#### Definition 2.4

*For a simplicial complex* *Δ* *having dimension* *d*, *its f* − *vector is a* *d* + 1-*tuple*, *defined as*:

$$\begin{array}{}{\displaystyle f(\mathit{\Delta})=({f}_{0},{f}_{1},\dots ,{f}_{d})}\end{array}$$

*where* *f*_{i} *denotes the number of* *i* − *dimensional faces of* *Δ*.

#### Definition 2.5 (Spanning Simplicial Complex)

*Let* *G*(*V*, *E*) *be a simple finite connected graph and* *s*(*G*) = {*E*_{1}, *E*_{2}, …, *E*_{t}} *be the edge*-*sets of all possible spanning trees of* *G*(*V*, *E*), *then we defined (in [**1**])* *a simplicial complex* *Δ*_{s}(*G*) *on* *E* *such that the facets of* *Δ*_{s}(*G*) *are precisely the elements of* *s*(*G*), *we call* *Δ*_{s}(*G*) *as the* *spanning simplicial complex of* *G*(*V*, *E*). *In other words*;

$$\begin{array}{}{\displaystyle {\mathit{\Delta}}_{s}(G)=\u3008{E}_{1},{E}_{2},\dots ,{E}_{t}\u3009.}\end{array}$$

For example, the spanning simplicial complex of the graph 𝓙_{2,3} given in Fig. 1 is:

*Δ*_{s}(𝓙_{2,3}) = 〈 {*e*_{11}, *e*_{21}, *e*_{31}, *e*_{12}, *e*_{22}, *e*_{32}}, {*e*_{11}, *e*_{21}, *e*_{31}, *e*_{12}, *e*_{22}, *e*_{33}}, {*e*_{11}, *e*_{21}, *e*_{31}, *e*_{12}, *e*_{23}, *e*_{32}}, {*e*_{11}, *e*_{21}, *e*_{31}, *e*_{12}, *e*_{23}, *e*_{33}}, {*e*_{11}, *e*_{21}, *e*_{31}, *e*_{13}, *e*_{22}, *e*_{32}}, {*e*_{11}, *e*_{21}, *e*_{31}, *e*_{13}, *e*_{22}, *e*_{33}}, {*e*_{11}, *e*_{21}, *e*_{31}, *e*_{13}, *e*_{23}, *e*_{32}}, {*e*_{11}, *e*_{21}, *e*_{31}, *e*_{13}, *e*_{23}, *e*_{33}}, {*e*_{21}, *e*_{31}, *e*_{32}, *e*_{33}, *e*_{12}, *e*_{22}}, {*e*_{21}, *e*_{31}, *e*_{32}, *e*_{33}, *e*_{12}, *e*_{23}}, {*e*_{21}, *e*_{31}, *e*_{32}, *e*_{33}, *e*_{13}, *e*_{22}}, {*e*_{21}, *e*_{31}, *e*_{32}, *e*_{33}, *e*_{13}, *e*_{23}}, {*e*_{21}, *e*_{31}, *e*_{12}, *e*_{13}, *e*_{33}, *e*_{22}}, {*e*_{21}, *e*_{31}, *e*_{12}, *e*_{13}, *e*_{33}, *e*_{23}}, {*e*_{21}, *e*_{31}, *e*_{12}, *e*_{13}, *e*_{32}, *e*_{22}}, {*e*_{21}, *e*_{31}, *e*_{12}, *e*_{13}, *e*_{32}, *e*_{23}}, {*e*_{11}, *e*_{31}, *e*_{12}, *e*_{13}, *e*_{22}, *e*_{32}}, {*e*_{11}, *e*_{31}, *e*_{12}, *e*_{13}, *e*_{22}, *e*_{33}}, {*e*_{11}, *e*_{31}, *e*_{12}, *e*_{13}, *e*_{23}, *e*_{32}}, {*e*_{11}, *e*_{31}, *e*_{12}, *e*_{13}, *e*_{23}, *e*_{33}}, {*e*_{11}, *e*_{31}, *e*_{22}, *e*_{23}, *e*_{13}, *e*_{32}}, {*e*_{11}, *e*_{31}, *e*_{22}, *e*_{23}, *e*_{13}, *e*_{33}}, {*e*_{11}, *e*_{31}, *e*_{22}, *e*_{23}, *e*_{12}, *e*_{32}}, {*e*_{11}, *e*_{31}, *e*_{22}, *e*_{23}, *e*_{12}, *e*_{33}}, {*e*_{11}, *e*_{21}, *e*_{23}, *e*_{22}, *e*_{32}, *e*_{12}}, {*e*_{11}, *e*_{21}, *e*_{23}, *e*_{22}, *e*_{32}, *e*_{13}}, {*e*_{11}, *e*_{21}, *e*_{23}, *e*_{22}, *e*_{33}, *e*_{12}}, {*e*_{11}, *e*_{21}, *e*_{23}, *e*_{22}, *e*_{33}, *e*_{13}}, {*e*_{11}, *e*_{21}, *e*_{32}, *e*_{33}, *e*_{22}, *e*_{12}}, {*e*_{11}, *e*_{21}, *e*_{32}, *e*_{33}, *e*_{22}, *e*_{13}}, {*e*_{11}, *e*_{21}, *e*_{32}, *e*_{33}, *e*_{23}, *e*_{12}}, {*e*_{11}, *e*_{21}, *e*_{32}, *e*_{33}, *e*_{23}, *e*_{13}}, {*e*_{11}, *e*_{13}, *e*_{22}, *e*_{23}, *e*_{32}, *e*_{33}}, {*e*_{11}, *e*_{12}, *e*_{22}, *e*_{23}, *e*_{32}, *e*_{33}}, {*e*_{11}, *e*_{12}, *e*_{13}, *e*_{23}, *e*_{32}, *e*_{33}}, {*e*_{11}, *e*_{12}, *e*_{13}, *e*_{22}, *e*_{32}, *e*_{33}}, {*e*_{11}, *e*_{12}, *e*_{13}, *e*_{22}, *e*_{23}, *e*_{33}}, {*e*_{11}, *e*_{12}, *e*_{13}, *e*_{22}, *e*_{23}, *e*_{32}}, {*e*_{21}, *e*_{13}, *e*_{22}, *e*_{23}, *e*_{32}, *e*_{33}}, {*e*_{21}, *e*_{12}, *e*_{22}, *e*_{23}, *e*_{32}, *e*_{33}}, {*e*_{21}, *e*_{12}, *e*_{13}, *e*_{23}, *e*_{32}, *e*_{33}}, {*e*_{21}, *e*_{12}, *e*_{13}, *e*_{22}, *e*_{32}, *e*_{33}}, {*e*_{21}, *e*_{12}, *e*_{13}, *e*_{22}, *e*_{23}, *e*_{33}}, {*e*_{21}, *e*_{12}, *e*_{13}, *e*_{22}, *e*_{23}, *e*_{32}}, {*e*_{31}, *e*_{13}, *e*_{22}, *e*_{23}, *e*_{32}, *e*_{33}}, {*e*_{31}, *e*_{12}, *e*_{22}, *e*_{23}, *e*_{32}, *e*_{33}}, {*e*_{31}, *e*_{12}, *e*_{13}, *e*_{23}, *e*_{32}, *e*_{33}}, {*e*_{31}, *e*_{12}, *e*_{13}, *e*_{22}, *e*_{32}, *e*_{33}}, {*e*_{31}, *e*_{12}, *e*_{13}, *e*_{22}, *e*_{23}, *e*_{33}}, {*e*_{31}, *e*_{12}, *e*_{13}, *e*_{22}, *e*_{23}, *e*_{32}}〉.

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