Star complements and connectivity in finite graphs

Let G be a finite graph with H as a star complement for an eigenvalue other than 0 or -1. Let κ(G), δ(G) denote respectively the vertex-connectivity and minimum degree of G. We prove that κ(G) is controlled by δ(G) and κ(H). In particular, for each k∈N there exists a smallest non-negative integer f(k) such that κ(G)⩾k whenever κ(H)⩾k and δ(G)⩾f(k). We show that f(1)=0, f(2)=2, f(3)=3, f(4)=5 and f(5)=7

A sufficient condition for the existence of fractional (g,f,n)(g,f,n)-critical covered graphs

In data transmission networks, the availability of data transmission is equivalent to the existence of the fractional factor of the corresponding graph which is generated by the network. Research on the existence of fractional factors under specific network structures can help scientists design and construct networks with high data transmission rates. A graph $G$ is called a fractional $(g,f)$-covered graph if for any $e\in E(G)$, $G$ admits a fractional $(g,f)$-factor covering $e$. A graph $G$ is called a fractional $(g,f,n)$-critical covered graph if after removing any $n$ vertices of $G$, the resulting graph of $G$ is a fractional $(g,f)$-covered graph. In this paper, we verify that if a graph $G$ of order $p$ satisfies $p\geq\frac{(a+b-1)(a+b-2)+(a+d)n+1}{a+d}$, $\delta(G)\geq\frac{(b-d-1)p+(a+d)n+a+b+1}{a+b-1}$ and $\delta(G)>\frac{(b-d-2)p+2\alpha(G)+(a+d)n+1}{a+b-2}$, then $G$ is a fractional $(g,f,n)$-critical covered graph, where $g,f:V(G)\rightarrow Z^{+}$ be two functions such that $a\leq g(x)\leq f(x)-d\leq b-d$ for all $x\in V(G)$, which is a generalization of Zhou's previous result [S. Zhou, Some new sufficient conditions for graphs to have fractional $k$-factors, International Journal of Computer Mathematics 88(3)(2011)484--490].Comment: 1

Newton flows for elliptic functions

Newton flows are dynamical systems generated by a continuous, desingularized Newton method for mappings from a Euclidean space to itself. We focus on the special case of meromorphic functions on the complex plane. Inspired by the analogy between the rational (complex) and the elliptic (i.e., doubly periodic meromorphic) functions, a theory on the class of so-called Elliptic Newton flows is developed. With respect to an appropriate topology on the set of all elliptic functions $f$ of fixed order $r$ ($\geq$ 2) we prove: For almost all functions $f$, the corresponding Newton flows are structurally stable i.e., topologically invariant under small perturbations of the zeros and poles for $f$ [genericity]. The phase portrait of a structurally stable elliptic Newton flow generates a connected, cellularly embedded, graph $G(f)$ on $T$ with $r$ vertices, 2$r$ edges and $r$ faces that fulfil certain combinatorial properties (Euler, Hall) on some of its subgraphs. The graph $G(f)$ determines the conjugacy class of the flow [characterization]. A connected, cellularly embedded toroidal graph $G$ with the above Euler and Hall properties, is called a Newton graph. Any Newton graph $G$ can be realized as the graph $G(f)$ of the structurally stable Newton flow for some function $f$ [classification]. This leads to: up till conjugacy between flows and(topological) equivalency between graphs, there is a 1-1 correspondence between the structurally stable Newton flows and Newton graphs, both with respect to the same order $r$ of the underlying functions $f$ [representation]. In particular, it follows that in case $r$ = 2, there is only one (up to conjugacy) structurally stabe elliptic Newton flow, whereas in case $r$ = 3, we find a list of nine graphs, determining all possibilities. Moreover, we pay attention to the so-called nuclear Newton flows of order $r$, and indicate how - by a bifurcation procedure - any structurally stable elliptic Newton flow of order $r$ can be obtained from such a nuclear flow. Finally, we show that the detection of elliptic Newton flows is possible in polynomial time. The proofs of the above results rely on Peixoto's characterization/classication theorems for structurally stable dynamical systems on compact 2-dimensional manifolds, Stiemke's theorem of the alternatives, Hall's theorem of distinct representatives, the Heter-Edmonds-Ringer rotation principle for embedded graphs, an existence theorem on gradient dynamical systems by Smale, and an interpretation of Newton flows as steady streams

THE MOSTAR INDEX OF FULLERENES IN TERMS OF AUTOMORPHISM GROUP

Let $G$ be a connected graph. For an edge $e=uv\in E(G)$, suppose $n(u)$ and $n(v)$ are respectively, the number of vertices of $G$ lying closer to vertex $u$ than to vertex $v$ and the number of vertices of $G$ lying closer to vertex $v$ than to vertex $u$. The Mostar index is a topological index which is defined as $Mo(G)=\sum_{e\in E(G)}f(e)$, where $f(e) = |n(u)-n(v)|$. In this paper, we will compute the Mostar index of a family of fullerene graphs in terms of the automorphism group. 

On the orthogonal rank of Cayley graphs and impossibility of quantum round elimination

After Bob sends Alice a bit, she responds with a lengthy reply. At the cost of a factor of two in the total communication, Alice could just as well have given the two possible replies without listening and have Bob select which applies to him. Motivated by a conjecture stating that this form of "round elimination" is impossible in exact quantum communication complexity, we study the orthogonal rank and a symmetric variant thereof for a certain family of Cayley graphs. The orthogonal rank of a graph is the smallest number $d$ for which one can label each vertex with a nonzero $d$-dimensional complex vector such that adjacent vertices receive orthogonal vectors. We show an exp$(n)$ lower bound on the orthogonal rank of the graph on $\{0,1\}^n$ in which two strings are adjacent if they have Hamming distance at least $n/2$. In combination with previous work, this implies an affirmative answer to the above conjecture.Comment: 13 page