A transitive signature scheme allows a signer to publish a graph in an authenticated and cost-saving manner. The resulting authenticated graph is indeed the transitive closure of the graph constructed by edges which are explicitly signed by the signer. A property of the transitive signature scheme enables such scenario is called composability. Composability means that by knowing signatures on t...

let $g$ be a simple graph of order $n$ and size $m$.the edge covering of $g$ is a set of edges such that every vertex of $g$ is incident to at least one edge of the set. the edge cover polynomial of $g$ is the polynomial$e(g,x)=sum_{i=rho(g)}^{m} e(g,i) x^{i}$,where $e(g,i)$ is the number of edge coverings of $g$ of size $i$, and$rho(g)$ is the edge covering number of $g$. in this paper we stud...

let g=(v,e) be a simple graph. an edge labeling f:e to {0,1} induces a vertex labeling f^+:v to z_2 defined by $f^+(v)equiv sumlimits_{uvin e} f(uv)pmod{2}$ for each $v in v$, where z_2={0,1} is the additive group of order 2. for $iin{0,1}$, let e_f(i)=|f^{-1}(i)| and v_f(i)=|(f^+)^{-1}(i)|. a labeling f is called edge-friendly if $|e_f(1)-e_f(0)|le 1$. i_f(g)=v_f(1)-v_f(0) is called the edge-f...

A fullerene graph is a cubic 3-connected plane graph with (exactly 12) pentagonal faces and hexagonal faces. Let Fn be a fullerene graph with n vertices. By the Euler formula one can see that Fn has 12 pentagonal and n/2 – 10 hexagonal faces. Let G = (V, E) be a connected graph with the vertices set V = V(G) and the edges set E = E(G), without loops and multiple edges. The distance d(x,y) betwe...

We investigate subsets of the symmetric group with structure similar to that of a graph. The “trees” of these subsets then lead to minimal highly symmetric generating sets of the symmetric group. We show that there exist generating sets among these with edge-transitive Cayley graphs and investigate them in relation to the Lovasz conjecture.

We consider the problem of finding a transitive orientation of a comparability graph, such that the edge set of its covering graph contains a given subset of edges. We propose a solution which employs the classical technique of modular tree decomposition. The method leads to a polynomial time algorithm to construct such an orientation or report that it does not exist. © 2006 Elsevier B.V. All r...

A graph G is said to be hyper-connected if the removal of every minimum cut creates exactly two connected components, one of which is an isolated vertex. In this paper, we first generalize the concept of hyper-connected graphs to that of semi-hyper-connected graphs: a graph G is called semi-hyper-connected if the removal of every minimum cut of G creates exactly two components. Then we characte...

We show that in any graph, the average length of a flow path in an electrical flow between the endpoints of a random edge is O(log n). This is a consequence of a more general result which shows that the spectral norm of the entrywise absolute value of the transfer impedance matrix of a graph is O(log n). This result implies a simple oblivious routing scheme based on electrical flows in the case...

Many large graphs can be constructed from existing smaller graphs by using graph operations, for example, the Cartesian product and the lexicographic product. Many properties of such large graphs are closely related to those of the corresponding smaller ones. In this short note, we give some properties of the lexicographic products of vertextransitive and of edge-transitive graphs. In particula...

Shortest transitive edge relation presents an alternative to transitive closure relation for answering a class of complex biological queries that requires connectivity information but not the full transitive closure. A biological querie such “ find each downstream protein interacting with glutamine synthetase using five or fewer interactions ” does not require transitive closure. Shortest trans...