A graph is symmetric if its automorphism group acts transitively on the set of arcs of the graph. In this paper, we classify tetravalent symmetric graphs of order 9p for each prime p.

Let E be a finite set, R be the set of real numbers and f:2E-+R be a symmetric submodular function. The pair (E,f) is called a symmetric submodular system. We examine the structures of symmetric submodular systems and provide a decomposition theory of symmetric submodular systems. The theory is a generalization of the decomposition theory of 2-connected graphs developed by Tutte and can be appl...

We solve a problem concerning a combinatorial design involving the transpositions in the symmetric group S2n. The problem admits equivalent formulations in terms of symmetric permutation matrices and 1-factorizations of complete graphs.

1. Definitions and a main result 2 1.1. Graphs and colorings . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2. Symmetric functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3. Chromatic symmetric functions . . . . . . . . . . . . . . . . . . . . . 5 1.4. Connected components . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.5. Circuits and broken circuits . . . . ...