Let C?(E) be the graph C?-algebra of a row-finite E. We give complete description vertex sets gauge-invariant regular ideals C?(E). It is shown that when E satisfies condition (L), are class which preserve (L) under quotients. That is, we show if then ideal J?C?(E) necessarily gauge-invariant. Further, ideal, it C?(E)?J?C?(F), where F (L).

Let $G$ be a non-abelian group. The non-commuting graph $Gamma_G$ of $G$ is defined as the graph whose vertex set is the non-central elements of $G$ and two vertices are joined if and only if they do not commute.In this paper we study some properties of $Gamma_G$ and introduce $n$-regular $AC$-groups. Also we then obtain a formula for Szeged index of $Gamma_G$ in terms of $n$, $|Z(G)|$ and $|G|...

A cellular embedding of an Eulerian digraph D into a closed surface is said to be directed if the boundary of each face is a directed closed walk in D. The directed genus polynomial of an Eulerian digraph D is the polynomial ΓD(x) = ∑ h≥0 gh(D)x h where gh(D) is the number of directed embeddings into the orientable surface Sh, of genus h, for h = 0, 1, . . . . The sequence {gh(D)|h ≥ 0}, which ...

An even-cycle decomposition of a graph G is a partition of E(G) into cycles of even length. Evidently, every Eulerian bipartite graph has an even-cycle decomposition. Seymour (1981) proved that every 2-connected loopless Eulerian planar graph with an even number of edges also admits an even-cycle decomposition. Later, Zhang (1994) generalized this to graphs with no K5-minor. Our main theorem gi...

the k-th semi total point graph of a graph g, , ‎is a graph‎ obtained from g by adding k vertices corresponding to each edge and‎ connecting them to the endpoints of edge considered‎. ‎in this paper‎, a formula for laplacian polynomial of in terms of‎ characteristic and laplacian polynomials of g is computed‎, ‎where is a connected regular graph‎.the kirchhoff index of is also computed‎.

A graph G = (V,E) is said to be weakly four-connected if G is 4-edgeconnected and G− x is 2-edge-connected for every x ∈ V . We prove that every weakly four-connected Eulerian graph has a 2-connected Eulerian orientation. This verifies a special case of a conjecture of A. Frank.