Cayley Sum Research Articles

Integral Cayley sum graphs and groups

Integral Cayley sum graphs and groups

Orthogonal Polarity Graphs and Sidon Sets

AbstractDetermining the maximum number of edges in an n‐vertex C4‐free graph is a well‐studied problem that dates back to a paper of Erdős from 1938. One of the most important families of C4‐free graphs are the Erdős‐Rényi orthogonal polarity graphs. We show that the Cayley sum graph constructed using a Bose‐Chowla Sidon set is isomorphic to a large induced subgraph of the Erdős‐Rényi orthogonal polarity graph. Using this isomorphism, we prove that the Petersen graph is a subgraph of every sufficiently large Erdős‐Rényi orthogonal polarity graph.

On Cayley Sum Graphs of Non-Abelian Groups

Let $$G$$G be a finite group and $$S$$S be a square-free normal subset of $$G$$G. The Cayley sum graph $$\mathrm{Cay}^+(G, S)$$Cay+(G,S) of $$G$$G with respect to $$S$$S is a simple graph whose vertex set is $$G$$G and two vertices $$g$$g and $$h$$h are joined by an edge if and only if $$g h \in S$$gh?S. In this paper, we obtain necessary and sufficient conditions on $$S$$S such that $$\mathrm{Cay}^+(G, S)$$Cay+(G,S) is connected. Also, we prove that if $$G$$G is a non-abelian group and $$|S|=3$$|S|=3, then $$\mathrm{Cay}^+(G, S)$$Cay+(G,S) is a connected integral graph if and only if $$G$$G is isomorphic to $$S_3$$S3, the symmetric group on $$3$$3 letters.

Cayley sum color and anti-circulant graphs

Let G be a finite group and α:G→R be a real-valued function on G. The Cayley sum color graph Cay+(G,α) is a complete directed graph with vertex set G where each arc (x,y)∈G×G is associated with a color α(xy). If α is the characteristic function on a subset S of G, then the Cayley sum graph Cay+(G,S) is obtained. The anti-circulant matrix associated to a vector v is an n×n matrix whose rows are given by iterations of the anti-shift operator acting on v. We note that a graph is a Cayley sum graph of a cyclic group if and only if it is an anti-circulant graph, a graph whose adjacency matrix is anti-circulant. In this paper, we obtain some results on the isomorphisms, connectivity and vertex transitivity about anti-circulant graphs. We find the spectrum of Cayley sum color graphs of abelian groups and as a result we compute the spectrum of real anti-circulant matrices.

CAYLEY SUM GRAPHS OF IDEALS OF A COMMUTATIVE RING

AbstractLet $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}R$ be a commutative ring, $I(R)$ be the set of all ideals of $R$ and $S$ be a subset of $I^*(R)=I(R)\setminus \{0\}$. We define a Cayley sum digraph of ideals of $R$, denoted by $\overrightarrow{\mathrm{Cay}}^+ (I(R),S)$, as a directed graph whose vertex set is the set $I(R)$ and, for every two distinct vertices $I$ and $J$, there is an arc from $I$ to $J$, denoted by $I\longrightarrow J$, whenever $I+K=J$, for some ideal $K $ in $S$. Also, the Cayley sum graph $ \mathrm{Cay}^+ (I(R), S)$ is an undirected graph whose vertex set is the set $I(R)$ and two distinct vertices $I$ and $J$ are adjacent whenever $I+K=J$ or $J+K=I$, for some ideal $K $ in $ S$. In this paper, we study some basic properties of the graphs $\overrightarrow{\mathrm{Cay}}^+ (I(R),S)$ and $ \mathrm{Cay}^+ (I(R), S)$ such as connectivity, girth and clique number. Moreover, we investigate the planarity, outerplanarity and ring graph of $ \mathrm{Cay}^+ (I(R), S)$ and also we provide some characterization for rings $R$ whose Cayley sum graphs have genus one.

On the domination and total domination numbers of Cayley sum graphs over Zn

Let G be a finite Abelian group and S be a subset of G. The Cayley sum graph Cay+(G, S) of G with respect to S is a graph whose vertex set is G and two vertices g and h are joined by an edge if and only if g + h є S. In this paper, we prove some basic facts on the domination and total domination numbers of Cayley sum graphs. Then, we find the sharp bounds for domination number of Cay+(Zn, S), where S = {1, 2,..., k} and
, k are positive integers with 1 ≤ k ≤ (n - 1)/2.

Cayley decompositions of lattice polytopes and upper bounds for h*-polynomials

We give an effective upper bound on the h-polynomial of a lattice polytope in terms of its degree and leading coefficient, confirming a conjecture of Batyrev. We deduce this bound as a consequence of a strong Cayley decomposition theorem which says, roughly speaking, that any lattice polytope with a large multiple that has no interior lattice points has a nontrivial decomposition as a Cayley sum of polytopes of smaller dimension. Polytopes with nontrivial Cayley decompositions correspond to projectivized sums of toric line bundles, and our approach is partially inspired by classification results of Fujita and others in algebraic geometry. In an appendix, we interpret our Cayley decomposition theorem in terms of adjunction theory for toric varieties.

Cayley sum graphs and eigenvalues of [formula omitted]-fullerenes

We determine the spectra of cubic plane graphs whose faces have sizes 3 and 6. Such graphs, “ ( 3 , 6 ) -fullerenes,” have been studied by chemists who are interested in their energy spectra. In particular we prove a conjecture of Fowler, which asserts that all their eigenvalues come in pairs of the form { λ , − λ } except for the four eigenvalues { 3 , − 1 , − 1 , − 1 } . We exhibit other families of graphs which are “spectrally nearly bipartite” in the sense that nearly all of their eigenvalues come in pairs { λ , − λ } . Our proof utilizes a geometric representation to recognize the algebraic structure of these graphs, which turn out to be examples of Cayley sum graphs.

Counting Sets With Small Sumset, And The Clique Number Of Random Cayley Graphs

Given a set A ⊂ ℤ/Nℤ we may form a Cayley sum graph G A on vertex set ℤ/Nℤ by joining i to j if and only if i+j ∈A. We investigate the extent to which performing this construction with a random set A simulates the generation of a random graph, proving that the clique number of G A is almost surely O(logN). This shows that Cayley sum graphs can furnish good examples of Ramsey graphs. To prove this result we must study the specific structure of set addition on ℤ/Nℤ. Indeed, we also show that the clique number of a random Cayley sum graph on Γ =(ℤ/2ℤ) n is almost surely not O(log |Γ|).