scholarly journals Resistance distance-based graph invariants and spanning trees of graphs derived from the strong prism of a graphs with given degree sequence

2020 ◽  
Author(s):  
Leilei Zhang
Keyword(s):  
Spanning Trees ◽  
Degree Sequence ◽  
Graph Invariants ◽  
Resistance Distance

A Note on the Kirchhoff and Additive Degree-Kirchhoff Indices of Graphs

2015 ◽  
Vol 70 (6) ◽  
pp. 459-463 ◽  
Author(s):  
Yujun Yang ◽  
Douglas J. Klein
Keyword(s):  
Upper Bound ◽  
Extremal Graphs ◽  
Graph Invariants ◽  
Kirchhoff Index ◽  
Resistance Distance

AbstractTwo resistance-distance-based graph invariants, namely, the Kirchhoff index and the additive degree-Kirchhoff index, are studied. A relation between them is established, with inequalities for the additive degree-Kirchhoff index arising via the Kirchhoff index along with minimum, maximum, and average degrees. Bounds for the Kirchhoff and additive degree-Kirchhoff indices are also determined, and extremal graphs are characterised. In addition, an upper bound for the additive degree-Kirchhoff index is established to improve a previously known result.

scholarly journals On relations between Kirchhoff index, Laplacian energy, Laplacian-energy-like invariant and degree deviation of graphs

Filomat ◽  
2020 ◽  
Vol 34 (3) ◽  
pp. 1025-1033
Author(s):  
Predrag Milosevic ◽  
Emina Milovanovic ◽  
Marjan Matejic ◽  
Igor Milovanovic
Keyword(s):  
Topological Index ◽  
Degree Sequence ◽  
Laplacian Matrix ◽  
Vertex Degree ◽  
Graph Invariants ◽  
Kirchhoff Index ◽  
Laplacian Eigenvalues ◽  
Laplacian Energy ◽  
Degree Deviation ◽  
Simple Connected Graph

Let G be a simple connected graph of order n and size m, vertex degree sequence d1 ? d2 ?...? dn > 0, and let ?1 ? ? 2 ? ... ? ?n-1 > ?n = 0 be the eigenvalues of its Laplacian matrix. Laplacian energy LE, Laplacian-energy-like invariant LEL and Kirchhoff index Kf, are graph invariants defined in terms of Laplacian eigenvalues. These are, respectively, defined as LE(G) = ?n,i=1 |?i-2m/n|, LEL(G) = ?n-1 i=1 ??i and Kf (G) = n ?n-1,i=1 1/?i. A vertex-degree-based topological index referred to as degree deviation is defined as S(G) = ?n,i=1 |di- 2m/n|. Relations between Kf and LE, Kf and LEL, as well as Kf and S are obtained.

scholarly journals Degree sequence realizations with given packing and covering of spanning trees

2015 ◽  
Vol 185 ◽  
pp. 113-118 ◽  
Author(s):  
Fengxia Liu ◽  
Zhao Zhang ◽  
Lai Hong-Jian ◽  
Meng Zhang
Keyword(s):  
Spanning Trees ◽  
Degree Sequence ◽  
Packing And Covering

scholarly journals On the Resistance Matrix of a Graph

10.37236/5295 ◽  
2016 ◽  
Vol 23 (1) ◽  
Author(s):  
Jiang Zhou ◽  
Zhongyu Wang ◽  
Changjiang Bu
Keyword(s):  
Regular Graph ◽  
Connected Graph ◽  
Spanning Trees ◽  
Regular Graphs ◽  
Resistance Distance ◽  
Strongly Regular ◽  
Resistance Matrix ◽  
Number Of Spanning Trees

Let $G$ be a connected graph of order $n$. The resistance matrix of $G$ is defined as $R_G=(r_{ij}(G))_{n\times n}$, where $r_{ij}(G)$ is the resistance distance between two vertices $i$ and $j$ in $G$. Eigenvalues of $R_G$ are called R-eigenvalues of $G$. If all row sums of $R_G$ are equal, then $G$ is called resistance-regular. For any connected graph $G$, we show that $R_G$ determines the structure of $G$ up to isomorphism. Moreover, the structure of $G$ or the number of spanning trees of $G$ is determined by partial entries of $R_G$ under certain conditions. We give some characterizations of resistance-regular graphs and graphs with few distinct R-eigenvalues. For a connected regular graph $G$ with diameter at least $2$, we show that $G$ is strongly regular if and only if there exist $c_1,c_2$ such that $r_{ij}(G)=c_1$ for any adjacent vertices $i,j\in V(G)$, and $r_{ij}(G)=c_2$ for any non-adjacent vertices $i,j\in V(G)$.

scholarly journals Subgraph counts for dense random graphs with specified degrees

2020 ◽  
pp. 1-38
Author(s):  
Catherine Greenhill ◽  
Mikhail Isaev ◽  
Brendan D. McKay
Keyword(s):  
Random Graph ◽  
Random Graphs ◽  
Spanning Trees ◽  
Degree Sequence ◽  
Complex Function ◽  
Random Permutation ◽  
Expected Number ◽  
Asymptotic Expressions ◽  
Number Of Spanning Trees

Abstract We prove two estimates for the expectation of the exponential of a complex function of a random permutation or subset. Using this theory, we find asymptotic expressions for the expected number of copies and induced copies of a given graph in a uniformly random graph with degree sequence(d 1 , …, d n ) as n→ ∞. We also determine the expected number of spanning trees in this model. The range of degrees covered includes d j = λn + O(n1/2+ε) for some λ bounded away from 0 and 1.

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