Chang, Y.-W., M.S. Jacobson, CL. Monma and D.B. West, Subtree and substar intersection numbers. Discrete Applied Mathematics 44 (1993) 205-220. We introduce the star number [free number] of a graph, being the minimum t such that G is the intersection graph of unions of t substars [subtrees] of a host tree. We study bounds on these parameters, compare them with interval number, and characterize ...

In 1981, Duffus, Gould, and Jacobson showed that every connected graph either has a Hamiltonian path, or contains claw (K1,3) net (a fixed six-vertex graph) as an induced subgraph. This implies subject to being connected, these two are the only minimal (under taking subgraphs) graphs with no path. Brousek (1998) characterized 2-connected, non-Hamiltonian do not contain We characterize 2-connect...

Let [Formula: see text] be a commutative ring with identity. The co-maximal ideal graph of text], denoted by is simple whose vertices are proper ideals which not contained in the Jacobson radical and two distinct adjacent if only text]. In this paper, we use Gallai’s Theorem concept strong resolving to compute metric dimension for graphs rings. Explicit formulae dimension, depending on whether ...

An old problem raised independently by Jacobson and Schönheim asks to determine the maximum s for which every graph with m edges contains a pair of edge-disjoint isomorphic subgraphs with s edges. In this paper we determine this maximum up to a constant factor. We show that every m-edge graph contains a pair of edge-disjoint isomorphic subgraphs with at least c(m logm) edges for some absolute c...

Given a graph H, a graphic sequence π is potentially H-graphic if there is some realization of π that contains H as a subgraph. In 1991, Erdős, Jacobson and Lehel posed the following question: Determine the minimum even integer σ(H,n) such that every n-term graphic sequence with sum at least σ(H,n) is potentially H-graphic. This problem can be viewed as a “potential” degree sequence relaxation ...

Gould, Jacobson and Lehel (Combinatorics, Graph Theory and Algorithms, Vol.I (1999) 451–460) considered a variation of the classical Turán-type extremal problems as follows: for any simple graph H, determine the smallest even integer σ(H, n) such that every n-term graphic sequence π = (d1, d2, . . . , dn) with term sum σ(π) = d1 + d2 + · · ·+ dn ≥ σ(H, n) has a realization G containing H as a s...

It is shown that an algebra Λ can be lifted with nilpotent Jacobson radical r = r(Λ) and has a generalized matrix unit {eii}I with each ēii in the center of Λ̄ = Λ/r iff Λ is isomorphic to a generalized path algebra with weak relations. Representations of the generalized path algebras are given. As a corollary, Λ is a finite algebra with non-zero unity element over perfect field k (e.g. a field ...

We extend the notion of the Leavitt path algebra of a graph E to include all directed graphs. We show how various ring-theoretic properties of these more general structures relate to the corresponding properties of Leavitt path algebras of row-finite graphs. Specifically, we identify those graphs for which the corresponding Leavitt path algebra is simple; purely infinite simple; exchange; and s...

Let H be a fixed graph. We say that graph G is -saturated if it has no subgraph isomorphic to , but the addition of any edge results in an -subgraph. The saturation number sat ( n ) minimum edges on vertices. Kászonyi and Tuza, 1986, gave general upper bound nontrivial lower remained elusive. In this paper we give prove asymptotically sharp (up additive constant) large class graphs. This includ...