An edge uv in a graph Γ is directionally 2-signed (or, (2, d)signed) by an ordered pair (a, b), a, b ∈ {+, −}, if the label l(uv) = (a, b) from u to v, and l(vu) = (b, a) from v to u. Directionally 2-signed graphs are equivalent to bidirected graphs, where each end of an edge has a sign. A bidirected graph implies a signed graph, where each edge has a sign. We extend a theorem of Sriraj and Sam...

The resistance distance between any two vertices of a connected graph is defined as the effective resistance between them in the electrical network constructed from the graph by replacing each edge with a (unit) resistor. The Kirchhoff index Kf(G) is the sum of resistance distances between all pairs of vertices in G. For a graph G, let R(G) be the graph obtained from G by adding a new vertex co...

The edge-partitioning problem presented in this paper is motivated by a problem in the design of SONET telecommunication networks. In a SONET ring, customer sites are connected via a ring of fiber-optic cable, each one sending, receiving, and relaying messages through a device called an add–drop-multiplexer (ADM). A bandwidth request, or demand, is given for every pair of sites. Each SONET ring...

the edge detour index polynomials were recently introduced for computing theedge detour indices. in this paper we nd relations among edge detour polynomials for the2-dimensional graph of tuc4c8(s) in a euclidean plane and tuc4c8(s) nanotorus.

Let G be a graph with two non adjacent vertices and G′ the graph constructed from G by adding an edge between them. It is known that the trace of Q′ is 2 plus the trace of Q, where Q and Q′ are the signless Laplacian matrices of G and G′ respectively. So, the sum of the Q′-eigenvalues of G′ is the sum of the the Qeigenvalues of G plus two. It is said that Q-spectral integral variation occurs wh...

A simultaneous embedding with fixed edges (SEFE) of two planar graphs R and B is a pair of plane drawings of R and B that coincide when restricted to the common vertices and edges of R and B. We show that whenever R and B admit a SEFE, they also admit a SEFE in which every edge is a polygonal curve with few bends and every pair of edges has few crossings. Specifically: (1) if R and B are trees ...

Fault tolerant measures have played an important role in the reliability of an interconnection network. Edge connectivity, restricted-edge-connectivity, extra-edge-connectivity and super-edge-connectivity of many well-known interconnection networks have been explored. In this paper, we study the 2-extra-edge connectivity of a special class of graphs G(G0,G1; M) proposed by Chen et al. [Appl. Ma...

Consider a directed graph G in which every edge has an associated real-valued distance and a real-valued weight. The weight of an undirected circuit of G is the sum of the weights of the edges, whereas the distance of an undirected circuit is the sum of the distances of the forward edges of the circuit minus the sum of the distances of the backward edges. A trivial circuit is a two-edge circuit...

A nowhere-zero 3-flow in a graph G is an assignment of a direction and a value of 1 or 2 to each edge of G such that, for each vertex v in G, the sum of the values of the edges with tail v equals the sum of the values of the edges with head v. Motivated by results about the region coloring of planar graphs, Tutte conjectured in 1966 that every 4-edge-connected graph has a nowhere-zero 3-flow. T...

Consider the complete n-vertex graph whose edge-lengths are independent exponentially distributed random variables. Simultaneously for each pair of vertices, put a constant flow between them along the shortest path. Each edge gets some random total flow. In the n→∞ limit we find explicitly the empirical distribution of these edge-flows, suitably normalized.