A Steinhaus matrix is a binary square matrix of size n which is symmetric, with diagonal of zeros, and whose upper-triangular coefficients satisfy ai,j = ai−1,j−1+ai−1,j for all 2 6 i < j 6 n. Steinhaus matrices are determined by their first row. A Steinhaus graph is a simple graph whose adjacency matrix is a Steinhaus matrix. We give a short new proof of a theorem, due to Dymacek, which states...

In this note we classify when a skew Schur function is a positive linear combination of power sum symmetric functions. We then use this to determine precisely when any scalar multiple of a skew Schur function is the chromatic symmetric function of some graph. As a consequence we prove that of the classical bases for symmetric functions only certain scalar multiples of the elementary symmetric f...

A graph Γ is symmetric if its automorphism group acts transitively on the arcs of Γ, and s-regular if its automorphism group acts regularly on the set of s-arcs of Γ. Tutte (1947, 1959) showed that every cubic finite symmetric cubic graph is s-regular for some s ≤ 5. Djokovič and Miller (1980) proved that there are seven types of arc-transitive group action on finite cubic graphs, characterised...

The graph metric of an undirected graph can be represented by a symmetric matrix in which each entry is the graph distance between the corresponding nodes, i.e., the shortest path distance between them. This article presents an improved all-pairs Dijkstra’s algorithm for computing the graph metric on an undirected weighted graph. Taking the advantage of the symmetric property of the graph metri...

The chromatic quasisymmetric function of a graph was introduced by Shareshian and Wachs as a refinement of Stanley’s chromatic symmetric function. An explicit combinatorial formula, conjectured by Shareshian and Wachs, expressing the chromatic quasisymmetric function of the incomparability graph of a natural unit interval order in terms of power sum symmetric functions, is proven. The proof use...

The symmetric traveling salesman problem (STSP) is a special case of ATSP in which the cost function is symmetric (i.e. c(u, v) = c(v, u)). Christofides in 1976 [6] discovered a 3/2approximation algorithm for STSP. No better approximation algorithm has since been found for the general symmetric metric case. However, polynomial-time approximation schemes have been found when the distances are sh...

The symmetric traveling salesman problem (STSP) is a special case of ATSP in which the cost function is symmetric (i.e. c(u, v) = c(v, u)). Christofides in 1976 [6] discovered a 3/2approximation algorithm for STSP. No better approximation algorithm has since been found for the general symmetric metric case. However, polynomial-time approximation schemes have been found when the distances are sh...

Two objects of fundamental importance in the study of graph polynomials are the poset of induced subgraphs of a graph and the lattice of its connected partitions. In my earlier paper I showed that several invariants of a graph can be computed from the isomorphism class of its poset of non-empty induced subgraphs, (that is, subgraphs themselves are not required). In this paper I will prove that ...

The paper describes a construction of abstract polytopes from Cayley graphs of symmetric groups. Given any connected graph G with p vertices and q edges, we associate with G a Cayley graph G(G) of the symmetric group Sp and then construct a vertex-transitive simple polytope of rank q, the graphicahedron, whose 1-skeleton (edge graph) is G(G). The graphicahedron of a graph G is a generalization ...

A graph is symmetric or arc-transitive if its automorphism group acts transitively on vertices, edges and arcs. Let p, q be odd primes with p, q ≥ 5 and X a q-valent symmetric graph of order 4p. In this paper, we proved that X K4p with 4p-1=q, X K2p,2p-2pK2 with 2p-1=q, the quotient graph of X is isomorphic to Kp,p and p=q, or K2p and 2p-1=q.