In this short paper, we present a solution to Gutman's problem on the characteristic polynomial of a bipartite graph.

Let D be a directed Eulerian multigraph, v be a vertex of D. We call the common value of id (v) and od(v) the degree of v, and simply denote it by dc. Xia introduced the concept of the T-transformation for directed Ealer tours and proved that any directed Euler tour (T)-transformation graph Eu(D) is connected. Zhang and Guo proved that Eu(D) is edge-Hamiltonian, i.e., any edge of Eu(D) is contained in a Hamilton cyclc of Eu(D). In this paper, we obtain a lower bound ∑Γ∈Q(dv-1)(dv- 2)/2 for the connectivity of Eu(D), where Q={v∈V(D)|du≥2}. Examples are given to show that this lower hound is in some sense best possible.

For a set C of cycles of a connected graph G we define T (G, C) as the graph with one vertex for each spanning tree of G, in which two trees R and S are adjacent if R U S contains exactly one cycle and this cycle lies in C. We give necessary conditions and sufficient conditions for T (G, C) to be connected.

Let G be an edge-colored graph. A heterochromatic path of G is such a path in which no two edges have the same color, de(v) denotes the color degree of a vertex v of G. In a previous paper, we showed that if dC(v)≥k for every vertex v of G, then G has a heterochromatic path of length at least [k+1/2]. It is easy to see that if k=1, 2, G has a heterochromatic path of length at least k. Saito conjectured that under the color degree condition G has a heterochromatic path of length at least [2k+1/3]. Even if this is true, no one knows if it is a best possible lower bound. Although we cannot prove Saito's conjecture, we can show in this paper that if 3≤k≤7, G has a heterochromatic path of length at least k-1, and if k≥8, G has a heterochromatic path of length at least [3k/5]+1. Actually, we can show" that for 1≤k≤5 any graph G under the color degree condition has a heterochromatic path of length at least k, with only one exceptional graph K4 for k=3, one exceptional graph for k=4 and three exceptional graphs for k=5, for which G has a heterochromatic path of length at least k-1. Our experience suggests us to conjecture that under the color degree condition G has a heterochromatic path of length at least k-1.

We give elementary recursive constructions of binary selforthogonal codes with dual distance four for all even lengths 12 and=8. Consequently, good quantum codes of minimum distance three and four for such length are obtained via Steane's construction and the CSS construction. Previously, such quantum codes were explicitly constructed only for a sparse set of lengths. Almost all of our quantum codes of minimum distance three are optimal or near optimal, and some of our minimum-distance four quantum codes are better than or comparable with those known before.