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It is shown that the line graph transformation

Because of its intrinsic interest, the line graph transformation

In this paper, a new topological invariance associated with the line graph transformation is found using the natural relationship between a Krausz decomposition of

The remainder of this paper is organized as follows: The relevant definitions and terminology are summarized in the next section. Required preliminary lemmas are provided in

A _{n}_{1} and _{2} of cardinality _{1} is adjacent to every vertex in _{2}, then _{m,n}_{1} is _{2} (denoted _{1} ≅ _{2}) if there is an adjacency preserving bijective map 𝜑: _{1}) → _{2}).

Associate with any non-empty graph _{1} and _{2} are non-trivial connected graphs—_{1}) ≅ _{2}) if, and only if, _{1} ≅ _{2} or {_{1},_{2}} is (up to isomorphism) the unordered pair {_{3},_{1,3}} [

A

If _{E }_{∈ }_{𝓔 }_{k}_{k}

Let {_{0},…, _{k}^{n}^{k}_{0},…, _{k}^{n}_{0},…,_{k}_{0} _{≤}_{ i} _{≤} _{k }_{i }a_{i}_{0} _{≤} _{i}_{≤}_{ k} λ_{i}_{0},…, _{k}^{ k}^{ k}_{0},…, _{k}_{i}_{i }L_{i}_{i }L_{i}

A

Two abstract complexes 𝓢 and 𝓣 are isomorphic if there is a bijection _{0},…, _{k}_{0}),…, _{k}

To each simplicial complex _{p}_{p+}_{1} : 𝒞_{p+}_{1}(_{p}_{p}_{p}_{p}_{p}_{p}^{th} homology group_{p}_{p}_{p}_{+1} and its ^{th} betti number b_{p}_{p}_{p}_{p}_{0}(

A cover of a simplicial complex _{α}_{α}_{α}_{α}_{α}_{α}_{0},…,_{n}_{∆} _{α}

The following lemmas are required to prove the main results in the next section. The first four are well known and are stated without proof for completeness.

∑_{0}_{ ≤}_{ p}_{ ≤}_{ m}^{p}_{p}_{0} _{≤}_{ p} _{≤}_{ m} (−1)^{p}b_{p}

_{0}(

The closure operation

_{m} : m

(_{m} Cl_{m}_{m} E_{m}

(

(

_{E }_{∈ }_{𝓔} _{i}, E_{j}, E_{k}_{i} of _{i}

(_{i}_{j}

(_{i}_{j}_{k}

_{i}_{j}_{i}_{j}_{i}_{j}_{k}_{m}_{m}_{i}_{j}_{k}

The terminology and results of the previous sections are now used to prove the following main results of this paper. In what follows, it is assumed that: (_{3} or _{1,3}; and (_{i}_{i}_{i}_{i}

_{i}_{j}_{3} or _{1,3}).

_{i}_{i}_{i}_{i}_{i}

∑_{1} _{≤} _{k}_{≤} _{𝝃} (^{k−}^{1}_{k}

_{k−}_{1}(_{k}

_{0}(_{0}(

∑_{1} _{≤} _{k}_{≤} _{𝝃} (−1)^{k−}^{1}_{k}

In order to illustrate the theory developed above, consider the non-trivial connected (4,4) graph _{1} ={a,b}, _{2} = {a,d}, _{3} = {b,c,d}, and _{4} = {c} as it hyperedges. Their closures are

_{1}) = {{a,b},{a},{b}},

_{2}) = {{a,d},{a},{d}},

_{3}) = {{b,c,d},{b,c},{b,d},{c,d},{b},{c},{d}},

and

_{4}) = {{c}},

so that

ℒ = {_{1}),_{2}),_{3}),_{4})}

and

It is clear that ℒ is an acyclic cover of the Krausz complex associated with

The line graph of a (4,4) graph and its Krausz decomposition.

The nerve

_{1}) ⋂ _{2}) ≠ Ø, _{1}) ⋂ _{3}) ≠ Ø, _{2}) ⋂ _{3}) ≠ Ø, _{3}) ⋂ _{4}) ≠ Ø

and

_{1}) ⋂ _{4}) = Ø, _{2}) ⋂ _{4}) = Ø

then the doubleton subsets {1,2}, {1,3}, {2,3}, and {3,4} are

The nerve of the acyclic cover of the Krausz complex associated with

Since _{3} = 1, _{2} = 5, _{1} = 4, and 𝝃 = 3, so that − as required by Corollary 1−

∑_{1 ≤ k ≤ 3}(-1)^{k-1}h_{k }= 4 − 5 + 1 = 0 =

It has been shown that a Krausz decomposition of the line graph of a graph

This work was supported by a grant from the Naval Surface Warfare Center Dahlgren Division’s In-house Laboratory Independent Research program.