In this paper, we consider the metric packing problem for the commodity graph of disjoint two triangles K3 +K3, which is dual to the multiflow feasibility problem for the commodity graph K3 + K3. We prove Karzanov’s conjecture concerning quarter-integral packings by certain bipartite metrics.

The aim of this article to follow the properties zero-divisor graph special idealization ring. We study wiener index zero-divisors some ring R(+)M and find clique number Γ(R(+)M) is ω (Γ(R(+)M)) = |M| − 1, where R an integral domain. also discuss when are Hamiltonian graph.

We completely describe the structure of irreducible integral flows on a signed graph by lifting them to the signed double covering graph. A (real-valued) flow (sometimes also called a circulation) on a graph or a signed graph (a graph with signed edges) is a real-valued function on oriented edges, f : ~ E → R, such that the net inflow to any vertex is zero. An integral flow is a flow whose valu...

The fractional and circular chromatic numbers are the two most studied non-integral refinements of number a graph. Starting from definition coloring base graph, which originated in work related to ergodic theory, we formalize notion gyrocoloring graph: vertices colored by translates single Borel set circle group, neighboring receive disjoint translates. corresponding gyrochromatic graph always ...

The simplicial rook graph SR(m,n) is the graph of which the vertices are the sequences of nonnegative integers of length m summing to n, where two such sequences are adjacent when they differ in precisely two places. We show that SR(m,n) has integral eigenvalues, and smallest eigenvalue s = max(−n,− ( m 2 ) ), and that this graph has a large part of its spectrum in common with the Johnson graph...

A (1, λ)-embedded graph is a graph that can be embedded on a surface with Euler characteristic λ so that each edge is crossed by at most one other edge. A graph G is called α-linear if there exists an integral constant β such that e(G′) ≤ αv(G′) + β for each G′ ⊆ G. In this paper, it is shown that every (1, λ)-embedded graph G is 4-linear for all possible λ, and is acyclicly edge-(3∆(G) + 70)-c...

For a given graph G, denote by A its adjacency matrix and F (t) = exp(iAt). We say that there exists a perfect state transfer (PST) in G if |F (τ)ab| = 1, for some vertices a, b and a positive real number τ . Such a property is very important for the modeling of quantum spin networks with nearest-neighbor couplings. We consider the existence of the perfect state transfer in integral circulant g...

Given a graph G = (V; E), the metric polytope S(G) is deened by the inequalities x(F) ? x(C n F) jFj ? 1 for F C; jFj odd ; C cycle of G, and 0 x e 1 for e 2 E. Optimization over S(G) provides an approximation for the max-cut problem. The graph G is called 1 d-integral if all the vertices of S(G) have their coordinates in f i d j 0 i dg. We prove that the class of 1 d-integral graphs is closed ...

In this paper, we developed some new formulae for exclusive sum number of the complete bipartite graph , n m K . Also we show that for     , 2 2 , 6 n n K E n K n    2 4, 2 6, 3 n n n          and 2 4 n   .

Suppose that G = (1/G, EG) is a planar graph embedded in the euclidean plane, that I, J, K, 0 are four of its faces (called holes in G), that sl, , s,, t, , , t, are vertices of G such that each pair {si, ti} belongs to the boundary of some of I,J, K, 0, and that the graph (I/G, EGu{ {.slr fl}, ___ ,{.r,, t,}}) is eulerian. We prove that if the multi(commodity)flow problem in G with unit demand...