The Vertex-Gutman Index of the Wheel Graphs

A labeling $f: E(G) \rightarrow \{1, -1\}$ of a graph G is called zero-M-cordial, if for each vertex v, the arithmetic sum of the labels occurrence with it is zero and $|e_{f}(-1) - e_{f}(1)| \leq 1$. A graph G is said to be Zero-M-cordial if a Zero-M-cordial label is given. Here the exploration of zero - M cordial labelings for deeds of paths, cycles, wheel and combining two wheel graphs, two Gear graphs, two Helm graphs. Here, also perceived that a zero-M-cordial labeling of a graph need not be a H-cordial labeling.

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On distance matrices of helm graphs obtained from wheel graphs with an even number of vertices

Linear Algebra and its Applications ◽

10.1016/j.laa.2021.03.008 ◽

2021 ◽

Vol 621 ◽

pp. 86-104

Author(s):

Shivani Goel

Keyword(s):

Wheel Graphs ◽

Distance Matrices

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A Kind Of Conditional Vertex Connectivity Of Cayley Graphs Generated By Wheel Graphs

The Computer Journal ◽

10.1093/comjnl/bxz077 ◽

2019 ◽

Vol 63 (9) ◽

pp. 1372-1384

Author(s):

Zuwen Luo ◽

Liqiong Xu

Keyword(s):

Fault Tolerance ◽

Connected Graph ◽

Cayley Graphs ◽

Network Structures ◽

Vertex Connectivity ◽

Wheel Graphs

Abstract Let $G=(V(G), E(G))$ be a connected graph. A subset $T \subseteq V(G)$ is called an $R^{k}$-vertex-cut, if $G-T$ is disconnected and each vertex in $V(G)-T$ has at least $k$ neighbors in $G-T$. The cardinality of a minimum $R^{k}$-vertex-cut is the $R^{k}$-vertex-connectivity of $G$ and is denoted by $\kappa ^{k}(G)$. $R^{k}$-vertex-connectivity is a new measure to study the fault tolerance of network structures beyond connectivity. In this paper, we study $R^{1}$-vertex-connectivity and $R^{2}$-vertex-connectivity of Cayley graphs generated by wheel graphs, which are denoted by $AW_{n}$, and show that $\kappa ^{1}(AW_{n})=4n-7$ for $n\geq 6$; $\kappa ^{2}(AW_{n})=6n-12$ for $n\geq 6$.

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The expected values for the Gutman index, Schultz index, multiplicative degree-Kirchhoff index and additive degree-Kirchhoff index of a random cyclooctatetraene chain

10.22541/au.163820385.52737069/v1 ◽

2021 ◽

Author(s):

Xianya Geng ◽

Jinfeng Qi ◽

Minjie Zhang

Keyword(s):

Kirchhoff Index ◽

Schultz Index ◽

Expected Values ◽

Gutman Index ◽

Analytical Expressions

In this paper, we mainly solve the explicit analytical expressions for the expected values of the Gutman index, Schultz index, multiplicative degree-Kirchhoff index and additive degree-Kirchhoff index of a random cyclooctatetraene chain with $n$ octagons. We also obtain the average values of these four indices with respect to the set of all these cyclooctatetraene chains.

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