The cubic graphs with finite cyclic vertex connectivity larger than girth

This paper considers commuting graphs over the semidihedral group SD8n. We compute their eigenvalues and obtain that these commuting graphs are not hyperenergetic for odd n≥15 or even n≥2. We further compute the Laplacian spectrum, the Laplacian energy and the number of spanning trees of the commuting graphs over SD8n. We also discuss vertex connectivity, planarity, and minimum disconnecting sets of these graphs and prove that these commuting graphs are not Hamiltonian.

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The genus of graphs associated with vector spaces

Journal of Algebra and Its Applications ◽

10.1142/s0219498820500863 ◽

2019 ◽

Vol 19 (05) ◽

pp. 2050086 ◽

Cited By ~ 1

Author(s):

T. Tamizh Chelvam ◽

K. Prabha Ananthi

Keyword(s):

Finite Field ◽

Vector Space ◽

Undirected Graph ◽

Vector Spaces ◽

Dimensional Vector ◽

Vertex Connectivity ◽

Dimensional Vector Space ◽

Nonzero Component ◽

Vertex Set ◽

Component Graph

Let [Formula: see text] be a k-dimensional vector space over a finite field [Formula: see text] with a basis [Formula: see text]. The nonzero component graph of [Formula: see text], denoted by [Formula: see text], is a simple undirected graph with vertex set as nonzero vectors of [Formula: see text] such that there is an edge between two distinct vertices [Formula: see text] if and only if there exists at least one [Formula: see text] along which both [Formula: see text] and [Formula: see text] have nonzero scalars. In this paper, we find the vertex connectivity and girth of [Formula: see text]. We also characterize all vector spaces [Formula: see text] for which [Formula: see text] has genus either 0 or 1 or 2.

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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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