W-graphs of representations of symmetric groups
When a line graph associated to annihilating-ideal graph of a lattice is planar or projective
Let be a finite lattice with a least element 0. is an annihilating-ideal graph of in which the vertex set is the set of all nontrivial ideals of , and two distinct vertices and are adjacent if and only if . We completely characterize all finite lattices whose line graph associated to an annihilating-ideal graph, denoted by , is a planar or projective graph.
When can the sum of th of the binomial coefficients have closed form.
When can you tile a box with translates of two given rectangular bricks?
When does the inverse have the same sign pattern as the transpose?
By a sign pattern (matrix) we mean an array whose entries are from the set . The sign patterns for which every real matrix with sign pattern has the property that its inverse has sign pattern are characterized. Sign patterns for which some real matrix with sign pattern has that property are investigated. Some fundamental results as well as constructions concerning such sign pattern matrices are provided. The relation between these sign patterns and the sign patterns of orthogonal matrices...
When is a 0-1 knapsack a matroid?
When is a poset isomorphic to the poset of connected induced subgraphs of a graph?
When is an Incomplete 3 × n Latin Rectangle Completable?
We use the concept of an availability matrix, introduced in Euler [7], to describe the family of all minimal incomplete 3 × n latin rectangles that are not completable. We also present a complete description of minimal incomplete such latin squares of order 4.
When is the algebra of multisymmetric polynomials generated by the elementary multisymmetric polynomials?
When is the orbit algebra of a group an integral domain ? Proof of a conjecture of P.J. Cameron
Cameron introduced the orbit algebra of a permutation group and conjectured that this algebra is an integral domain if and only if the group has no finite orbit. We prove that this conjecture holds and in fact that the age algebra of a relational structure R is an integral domain if and only if R is age-inexhaustible. We deduce these results from a combinatorial lemma asserting that if a product of two non-zero elements of a set algebra is zero then there is a finite common tranversal of their...
When structures are almost surely connected.
Where the typical set partitions meet and join.
Which Cayley graphs are integral?
Which chessboards have a closed knight's tour within the cube?
Which chessboards have a closed knight's tour within the rectangular prism?
Which directed graphs have a solution?
Why the characteristic polynomial factors.
Width compactness theorem and well-quasi-ordering infinite graphs
Wiener and vertex PI indices of the strong product of graphs
The Wiener index of a connected graph G, denoted by W(G), is defined as . Similarly, the hyper-Wiener index of a connected graph G, denoted by WW(G), is defined as . The vertex Padmakar-Ivan (vertex PI) index of a graph G is the sum over all edges uv of G of the number of vertices which are not equidistant from u and v. In this paper, the exact formulae for Wiener, hyper-Wiener and vertex PI indices of the strong product , where is the complete multipartite graph with partite sets of sizes...
Wiener index of generalized stars and their quadratic line graphs
The Wiener index, W, is the sum of distances between all pairs of vertices in a graph G. The quadratic line graph is defined as L(L(G)), where L(G) is the line graph of G. A generalized star S is a tree consisting of Δ ≥ 3 paths with the unique common endvertex. A relation between the Wiener index of S and of its quadratic graph is presented. It is shown that generalized stars having the property W(S) = W(L(L(S)) exist only for 4 ≤ Δ ≤ 6. Infinite families of generalized stars with this property...