Preface 1. Introduction 2. Graph operations and modifications 3. Spectrum and structure 4. Characterizations by spectra 5. Structure and one eigenvalue 6. Spectral techniques 7. Laplacians 8. Additional topics 9. Applications Appendix Bibliography Index of symbols Index.

An Introduction to the Theory of Graph Spectra: Spectral techniques

The enhanced power graph Pe(G) of a group G is a graph with vertex set G and two vertices are adjacent if they belong to the same cyclic subgroup. In this paper, we consider the minimum degree, ind...

On the enhanced power graph of a finite group

This paper deals with the vertex connectivity of enhanced power graphs of finite groups. We classify all abelian groups G such that the vertex connectivity of enhanced power graph of G is 1. We derive an upper bound for the vertex connectivity of the enhanced power graph of any general abelian group G. Also we completely characterize all abelian groups G, such that the proper enhanced power graph is connected. Moreover, we study some special class of non-abelian groups G such that the proper enhanced power graph is connected and we find their vertex connectivity.

On the Connectivity of Enhanced Power Graphs of Finite Groups

This paper deals with the vertex connectivity of enhanced power graph of finite group. We classify all abelian groups G such that vertex connectivity of enhanced power graph of G is 1. We derive an upper bound of vertex connectivity for the enhanced power graph of any general abelian group G. Also we completely characterize all abelian group G, such that the proper enhanced power graph is connected. Moreover, we study some special class of non-abelian group G such that the proper enhanced power graph is connected and we find their vertex connectivity.

On the connectivity of enhanced power graph of finite group

The commuting graph $$\varDelta (G)$$
 of a finite non-abelian group G is a simple graph with vertex set G, and two distinct vertices x, y are adjacent if $$xy = yx$$
 . In this paper, first we discuss some properties of $$\varDelta (G)$$
 . We determine the edge connectivity and the minimum degree of $$\varDelta (G)$$
 and prove that both are equal. Then, other graph invariants, namely: matching number, clique number, boundary vertex, of $$\varDelta (G)$$
 are studied. Also, we give necessary and sufficient condition on the group G such that the interior and center of $$\varDelta (G)$$
 are equal. Further, we investigate the commuting graph of the semidihedral group $$SD_{8n}$$
 . In this connection, we discuss various graph invariants of $$\varDelta (SD_{8n})$$
 including vertex connectivity, independence number, matching number and detour properties. We also obtain the Laplacian spectrum, metric dimension and resolving polynomial of $$\varDelta (SD_{8n})$$
 .

On the Commuting Graph of Semidihedral Group

The enhanced power graph &#x1d4ab;e(G) of a group G is a simple undirected graph with vertex set G and two distinct vertices x,y are adjacent if both x and y belongs to same cyclic subgroup of G. In this p...

On enhanced power graphs of certain groups

The cyclic graph $$\Gamma (S)$$ of a semigroup S is the simple graph whose vertex set is S, two element being adjacent if the subsemigroup generated by these two elements is monogenic. The purpose of this note is to prove that the chromatic number of $$\Gamma (S)$$ is at most countable. The present paper generalizes the results of Shitov (Graphs Comb 33(2):485–487, 2017) and the corresponding results on power graph and enhanced power graph of groups obtained by Aalipour et al. (Electron J Comb 24(3):#P3.16, 2017).

Chromatic Number of the Cyclic Graph of Infinite Semigroup

. The enhanced power graph of a ﬁnite group G is the simple undirected graph whose vertex set is G and two distinct vertices x,y are adjacent if x,y ∈ h z i for some z ∈ G . An L (2 , 1)-labeling of graph Γ is an integer labeling of V (Γ) such that adjacent vertices have labels that diﬀer by at least 2 and vertices distance 2 apart have labels that diﬀer by at least 1. The λ -number of Γ, denoted by λ (Γ), is the minimum range over all L (2 , 1)-labelings. In this article, we study the lambda number of the enhanced power graph P E ( G ) of the group G . This paper extends the corresponding results, obtained in [X. Ma, M. Feng, and K. Wang. Lambda number of the power graph of a ﬁnite group. J. Algebraic Combin. , 53(3):743–754, 2021], of the lambda number of power graphs to enhanced power graphs. Moreover, for a non-trivial simple group G of order n , we prove that λ ( P E ( G )) = n if and only if G is not a cyclic group of order n ≥ 3. Finally, we compute the exact value of λ ( P E ( G )) if G is a ﬁnite nilpotent group.

Sandeep Dalal

Papers

On the enhanced power graph of a finite group

On the Commuting Graph of Semidihedral Group

On enhanced power graphs of certain groups

Chromatic Number of the Cyclic Graph of Infinite Semigroup

Lambda Number of the enhanced power graph of a finite group