Resistance distance-based graph invariants and spanning trees of graphs derived from the strong prism of a star

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.

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Resistance distance-based graph invariants of subdivisions and triangulations of graphs

Discrete Applied Mathematics ◽

10.1016/j.dam.2014.08.039 ◽

2015 ◽

Vol 181 ◽

pp. 260-274 ◽

Cited By ~ 26

Author(s):

Yujun Yang ◽

Douglas J. Klein

Keyword(s):

Graph Invariants ◽

Resistance Distance

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On the Resistance Matrix of a Graph

The Electronic Journal of Combinatorics ◽

10.37236/5295 ◽

2016 ◽

Vol 23 (1) ◽

Cited By ~ 7

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 $i,j\in V(G)$.

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