Let $G$ be a non-trivial group, $S\subseteq G\setminus \{1\}$ and $S=S^{-1}:=\{s^{-1} \;|\; s\in S\}$. The Cayley graph of $G$ denoted by $\Gamma(S:G)$ is a graph with vertex set $G$ and two vertices $a$ and $b$ are adjacent if $ab^{-1}\in S$. A graph is called integral, if its adjacency eigenvalues are integers. In this paper we determine all connected cubic integral Cayley graphs. We also introduce some infinite families of connected integral Cayley graphs.
A graph is called integral, if its adjacency eigenvalues are integers. In this paper we determine integral quartic Cayley graphs on finite abelian groups. As a side result we show that there are exactly $27$ connected integral Cayley graphs up to $11$ vertices.
Let R be a non-commutative ring. The commuting graph of R denoted by Γ(R), is a graph with vertex set R\Z(R) and two vertices a and b are adjacent if ab = ba. It has been shown that the diameter of Γ(R)c is less than 3. For a finite ring R we show that the diameter of Γ(R)c is one if and only if R is the non-commutative ring on 4 elements. Also we characterize all rings where the complements of their commuting graphs are planar. Moreover, we identify the commuting graphs of rings of order pi for i = 2, 3 and prime number p.
Let G be a non-abelian group. The non-commuting graph of group G, shown by Γ G , is a graph with the vertex set G \ Z(G), where Z(G) is the center of group G. Also two distinct vertices of a and b are adjacent whenever ab = ba. A set S ⊆ V (Γ) of vertices in a graph Γ is a dominating set if every vertex v ∈ V (Γ) is an element of S or adjacent to an element of S. The domination number of a graph Γ denoted by γ(Γ), is the minimum size of a dominating set of Γ. Here, we study some properties of the non-commuting graph of some finite groups. In this paper, we show that γ(Γ G) < |G|−|Z(G)| 2. Also we charactrize all of groups G of order n with t = |Z(G)|, in which γ(Γ G) + γ(Γ G) ∈ {n − t + 1, n − t, n − t − 1, n − t − 2}.
Abstract. A graph is said to be determined by its signless Laplacian spectrum if there is no other non-isomorphic graph with the same spectrum. In this paper, it is shown that each starlike tree with maximum degree 4 is determined by its signless Laplacian spectrum.
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