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A complete oriented graph (Skiena 1990, p. 175), i.e., a graph in which every pair of nodes is connected by a single uniquely directed edge. The first and second 3-node ...

If the rank polynomial R(x,y) of a graph G is given by sumrho_(rs)x^ry^s, then rho_(rs) is the number of subgraphs of G with rank r and co-rank s, and the matrix (rho_(rs)) ...

The ABC (atom-bond connectivity) matrix A_(ABC) of a simple graph is a weighted adjacency matrix with weight f(d_i,d_j)=sqrt((d_i+d_j-2)/(d_id_j)), (1) where d_i are the ...

The Sombor matrix A_(Sombor) of a simple graph is a weighted adjacency matrix with weight f(d_i,d_j)=sqrt(d_i^2+d_j^2), (1) where d_i are the vertex degrees of the graph. In ...

A weighted adjacency matrix A_f of a simple graph is defined for a real positive symmetric function f(d_i,d_j) on the vertex degrees d_i of a graph as ...

Let I(G) denote the set of all independent sets of vertices of a graph G, and let I(G,u) denote the independent sets of G that contain the vertex u. A fractional coloring of ...

Given a complete graph K_n which is two-colored, the number of forced monochromatic triangles is at least {1/3u(u-1)(u-2) for n=2u; 2/3u(u-1)(4u+1) for n=4u+1; ...

Betti numbers are topological objects which were proved to be invariants by Poincaré, and used by him to extend the polyhedral formula to higher dimensional spaces. ...

A formula satisfied by all Hamiltonian cycles with n nodes. Let f_j be the number of regions inside the circuit with j sides, and let g_j be the number of regions outside the ...

The Egyptian Mathematical Leather Roll (EMLR), dates to the Middle Kingdom, and was purchased in Egypt in 1858 by Henry Rhind, near the time when the Rhind papyrus was ...