Clique-inserted-graphs and spectral dynamics of clique-inserting

Zhang, Fuji; Chen, Yi Chiuan; Chen, Zhibo

doi:10.1016/j.jmaa.2008.08.036

Cited by 30 publications

(26 citation statements)

References 23 publications

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“…By Theorem or Corollary , it is easy to obtain the following known results after simple analyses and calculations: Let G be a r ‐regular graph. Then (1)

t false(L (G) false) = 2^{m - n + 1} r^{m - n - 1} t false(G false)

; (2)

t false(L (S_{1} false(G false)) false) = r^{m - n - 1} {(r + 2)}^{m - n + 1} t false(G false)

, see Corollary 2.2 of . Let

R (G)

be the graph obtained from G by attaching

Δ - d (v)

pendent edges to each vertex v of G , where Δ is the maximum degree of G . Then (1)

t false(L (R false(G false)) false) = 2^{m - n + 1} {normalΔ}^{n Δ - m - n - 1} t false(G false)

, see Theorem 1.3 of . (2)

t false(L (S false(R (G) false)) false) = {normalΔ}^{n Δ - m - n - 1} {(Δ + 2)}^{m - n + 1} t false(G false)

. A bipartite graph G with bipartition

(V_{1}, V_{2})

is called

…”

Section: Main Results and Its Proofmentioning

confidence: 99%

“…The number of spanning trees of line graphs and its connection to the number of spanning trees in the original graphs have been studied since 1960s via matrix manipulations and the Matrix‐Tree Theorem . Many results have been obtained, for example, [, , , ]. Recently, by an elaborate and complicated combinatorial analysis, Dong and Yan in proved the following result:

\begin{matrix} true \\ right t false(L (S_{k} false(G false)) false) = \prod_{v \in V} d {(v)}^{d (v) - 2} \times \sum_{E^{'} \subseteq E} t false(G [E^{'}] false) k^{false| E^{'} false| - false| V false| + 1} \prod_{e \in E ∖ E^{'}} [d {false(u_{e} false)}^{- 1} + d {false(v_{e} false)}^{- 1}] . \end{matrix}

In the case of

k = 0

, Dong and Yan showed

0true t (L false(G false)) = \prod_{v \in V} d {false(v false)}^{d false(v false) - 2} \sum_{T \subseteq scriptT false(G false)} \prod_{e \in E ∖ E false(T false)} ([] d {(u_{e})}^{- 1} + d {(v_{e})}^{- 1}) .

…”

Section: Introductionmentioning

confidence: 99%

“…The number of spanning trees of line graphs and its connection to the number of spanning trees in the original graphs have been studied since 1960s via matrix manipulations and the Matrix-Tree Theorem [10]. Many results have been obtained, for example, [2,3,9,[17][18][19]. Recently, by an elaborate and complicated combinatorial analysis, Dong and Yan in [4] proved the following result:…”

mentioning

confidence: 99%

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A simple formula for the number of spanning trees of line graphs

Gong

2017

Journal of Graph Theory

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Suppose G=(V,E) is a loopless graph and Skfalse(Gfalse) is the graph obtained from G by subdividing each of its edges k (k≥0) times. Let T(G) be the set of all spanning trees of G, L(Skfalse(Gfalse)) be the line graph of the graph Skfalse(Gfalse) and t(Lfalse(Sk(G)false)) be the number of spanning trees of L(Skfalse(Gfalse)). By using techniques from electrical networks, we first obtain the following simple formula: truerightleftt(Lfalse(Sk(G)false))=1∏v∈Vd2false(vfalse)rightleft×∑T∈T(G)∏e=xy∈E(T)dfalse(xfalse)dfalse(yfalse)rightleft×∏e=uv∈E∖E(T)false[d(u)+kd(u)d(v)+d(v)false].Then we find it is in fact equivalent to a complicated formula obtained recently using combinatorial techniques in [F. M. Dong and W. G. Yan, Expression for the number of spanning trees of line graphs of arbitrary connected graphs, J. Graph Theory. 85 (2017) 74–93].

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“…By Theorem or Corollary , it is easy to obtain the following known results after simple analyses and calculations: Let G be a r ‐regular graph. Then (1)

t false(L (G) false) = 2^{m - n + 1} r^{m - n - 1} t false(G false)

; (2)

t false(L (S_{1} false(G false)) false) = r^{m - n - 1} {(r + 2)}^{m - n + 1} t false(G false)

, see Corollary 2.2 of . Let

R (G)

be the graph obtained from G by attaching

Δ - d (v)

pendent edges to each vertex v of G , where Δ is the maximum degree of G . Then (1)

t false(L (R false(G false)) false) = 2^{m - n + 1} {normalΔ}^{n Δ - m - n - 1} t false(G false)

, see Theorem 1.3 of . (2)

t false(L (S false(R (G) false)) false) = {normalΔ}^{n Δ - m - n - 1} {(Δ + 2)}^{m - n + 1} t false(G false)

. A bipartite graph G with bipartition

(V_{1}, V_{2})

is called

…”

Section: Main Results and Its Proofmentioning

confidence: 99%

\begin{matrix} true \\ right t false(L (S_{k} false(G false)) false) = \prod_{v \in V} d {(v)}^{d (v) - 2} \times \sum_{E^{'} \subseteq E} t false(G [E^{'}] false) k^{false| E^{'} false| - false| V false| + 1} \prod_{e \in E ∖ E^{'}} [d {false(u_{e} false)}^{- 1} + d {false(v_{e} false)}^{- 1}] . \end{matrix}

In the case of

k = 0

, Dong and Yan showed

0true t (L false(G false)) = \prod_{v \in V} d {false(v false)}^{d false(v false) - 2} \sum_{T \subseteq scriptT false(G false)} \prod_{e \in E ∖ E false(T false)} ([] d {(u_{e})}^{- 1} + d {(v_{e})}^{- 1}) .

…”

Section: Introductionmentioning

confidence: 99%

“…The number of spanning trees of line graphs and its connection to the number of spanning trees in the original graphs have been studied since 1960s via matrix manipulations and the Matrix-Tree Theorem [10]. Many results have been obtained, for example, [2,3,9,[17][18][19]. Recently, by an elaborate and complicated combinatorial analysis, Dong and Yan in [4] proved the following result:…”

mentioning

confidence: 99%

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A simple formula for the number of spanning trees of line graphs

Gong

2017

Journal of Graph Theory

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“…A paraline graph, denoted by ( ), is defined as a line graph of the subdivision graph ( ) (the subdivision graph ( ) of a graph is the graph obtained from by inserting a vertex to every edge of ) of (e.g., see Figure 1). The concept of the paraline graph (or clique-inserted graph [32]) of a graph was first introduced in [25], where the author obtained the spectrum of the paraline graph of a regular graph with infinite number of vertices in terms of the spectrum of .…”

Section: Lemma 7 Let Be An ( )-Semiregular Graph With Vertices Thenmentioning

confidence: 99%

Bounds for Incidence Energy of Some Graphs

Wang

Yang

2013

Journal of Applied Mathematics

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“…Such a relation was first found by Vahovskii , then by Kelmans and was rediscovered by Cvetković, Doob, and Sachs for regular graphs. They showed that if G is a k ‐regular graph of order n and size m , then

t false(L (G) false) = k^{m - n - 1} 2^{m - n + 1} t false(G false) .

The first result on the relation between

t (G)

and

t (L (S (G)))

was found by Zhang, Chen, and Chen . They proved that if G is k ‐regular, then

t false(L (S false(G false)) false) = k^{m - n - 1} {(k + 2)}^{m - n + 1} t false(G false) .

Yan recently generalized the result of .…”

Section: Introductionmentioning

confidence: 99%

Expression for the Number of Spanning Trees of Line Graphs of Arbitrary Connected Graphs

Dong

Yan

2016

Journal of Graph Theory

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For any graph G, let t(G) be the number of spanning trees of G, L(G) be the line graph of G, and for any nonnegative integer r, Srfalse(Gfalse) be the graph obtained from G by replacing each edge e by a path of length r+1 connecting the two ends of e. In this article, we obtain an expression for t(Lfalse(Sr(G)false)) in terms of spanning trees of G by a combinatorial approach. This result generalizes some known results on the relation between t(Lfalse(Sr(G)false)) and t(G) and gives an explicit expression tfalse(L(Srfalse(Gfalse))false)=km+s−n−1(rk+2)m−n+1tfalse(Gfalse) if G is of order n+s and size m+s in which s vertices are of degree 1 and the others are of degree k. Thus we prove a conjecture on t(Lfalse(S1(G)false)) for such a graph G.

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Clique-inserted-graphs and spectral dynamics of clique-inserting

Cited by 30 publications

References 23 publications

A simple formula for the number of spanning trees of line graphs

A simple formula for the number of spanning trees of line graphs

Bounds for Incidence Energy of Some Graphs

Expression for the Number of Spanning Trees of Line Graphs of Arbitrary Connected Graphs

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