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Property A is a form of weak amenability for groups and metric spaces introduced as an approach to the famous Novikov higher signature conjecture, one of the most important unsolved problems in topology. We show that property A can be reduced to a sequence of linear programming optimization problems on finite graphs. We explore the dual problems, which turn out to have interesting interpretations as combinatorial problems concerning the maximum total supply of flows on a network. Using isoperimetric inequalities, we relate the dual problems to the Cheeger constant of the graph and explore the role played by symmetry of a graph to obtain a striking characterization of the difference between an expander and a graph without property A. Property A turns out to be a new measure of connectivity of a graph that is relevant to graph theory. The dual linear problems can be solved using a variety of methods, which we demonstrate on several enlightening examples. As a demonstration of the power of this linear programming approach we give elegant proofs of theorems of Nowak and Willett about graphs without property A.
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For a graph $G= (V,E)$, a double Roman dominating function (DRDF) is a function $f : V to {0,1,2,3}$ having the property that if $f (v) = 0$, then vertex $v$ must have at least two neighbors assigned $2$ under $f$ or {at least} one neighbor $u$ with $f (u) = 3$, and if $f (v) = 1$, then vertex $v$ must have at least one neighbor $u$ with $f (u) ge 2$. In this paper, we consider the double Roman domination problem, which is an optimization problem of finding the DRDF $f$ such that $sum_{vin V} f (v)$ is minimum. We propose {five integer linear programming (ILP) formulations and one mixed integer linear programming formulation with polynomial number of constraints for this problem. Some additional valid inequalities and bounds are also proposed for some of these formulations.} Further, we prove that {the first four models indeed solve the double Roman domination problem, and the last two models} are equivalent to the others regardless of the variable relaxation or usage of a smaller number of constraints and variables. Additionally, we use one ILP formulation to give an $H(2(Delta+1))$-approximation algorithm. All proposed formulations and approximation algorithm are evaluated on randomly generated graphs to compare the performance.
In this paper we study finite groups which have Cayley isomorphism property with respect to Cayley maps, CIM-groups for a brief. We show that the structure of the CIM-groups is very restricted. It is described in Theorem~ref{111015a} where a short list of possible candidates for CIM-groups is given. Theorem~ref{111015c} provides concrete examples of infinite series of CIM-groups.
For a graph $G,$ the set $D subseteq V(G)$ is a porous exponential dominating set if $1 le sum_{d in D} left( 2 right)^{1-dist(d,v)}$ for every $v in V(G),$ where $dist(d,v)$ denotes the length of the shortest $dv$ path. The porous exponential dominating number of $G,$ denoted $gamma_e^*(G),$ is the minimum cardinality of a porous exponential dominating set. For any graph $G,$ a technique is derived to determine a lower bound for $gamma_e^*(G).$ Specifically for a grid graph $H,$ linear programing is used to sharpen bound found through the lower bound technique. Lower and upper bounds are determined for the porous exponential domination number of the King Grid $mathcal{K_n},$ the Slant Grid $mathcal{S_n},$ and the $n$-dimensional hypercube $Q_n.$
A Cayley (di)graph $Cay(G,S)$ of a group $G$ with respect to $S$ is said to be normal if the right regular representation of $G$ is normal in the automorphism group of $Cay(G,S)$, and is called a CI-(di)graph if there is $alphain Aut(G)$ such that $S^alpha=T$, whenever $Cay(G,S)cong Cay(G,T)$ for a Cayley (di)graph $Cay(G,T)$. A finite group $G$ is called a DCI-group or a NDCI-group if all Cayley digraphs or normal Cayley digraphs of $G$ are CI-digraphs, and is called a CI-group or a NCI-group if all Cayley graphs or normal Cayley graphs of $G$ are CI-graphs, respectively. Motivated by a conjecture proposed by Adam in 1967, CI-groups and DCI-groups have been actively studied during the last fifty years by many researchers in algebraic graph theory. It takes about thirty years to obtain the classification of cyclic CI-groups and DCI-groups, and recently, the first two authors, among others, classified cyclic NCI-groups and NDCI-groups. Even though there are many partial results on dihedral CI-groups and DCI-groups, their classification is still elusive. In this paper, we prove that a dihedral group of order $2n$ is a NCI-group or a NDCI-group if and only if $n=2,4$ or $n$ is odd. As a direct consequence, we have that if a dihedral group $D_{2n}$ of order $2n$ is a DCI-group then $n=2$ or $n$ is odd-square-free, and that if $D_{2n}$ is a CI-group then $n=2,9$ or $n$ is odd-square-free, throwing some new light on classification of dihedral CI-groups and DCI-groups.
Delsarte, Goethals, and Seidel (1977) used the linear programming method in order to find bounds for the size of spherical codes endowed with prescribed inner products between distinct points in the code. In this paper, we develop the linear programming method to obtain bounds for the number of vertices of connected regular graphs endowed with given distinct eigenvalues. This method is proved by some dual technique of the spherical case, motivated from the theory of association scheme. As an application of this bound, we prove that a connected $k$-regular graph satisfying $g>2d-1$ has the minimum second-largest eigenvalue of all $k$-regular graphs of the same size, where $d$ is the number of distinct non-trivial eigenvalues, and $g$ is the girth. The known graphs satisfying $g>2d-1$ are Moore graphs, incidence graphs of regular generalized polygons of order $(s,s)$, triangle-free strongly regular graphs, and the odd graph of degree $4$.
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