For a graph G, the Merrifield-Simmons index i(G) is defined as the total number of independent sets of the graph G. Let G(n, l, k) be the class of unicyclic graphs on n vertices with girth and pendent vertices being resp. l and k. In this paper, we characterize the unique unicyclic graph possessing prescribed girth and pendent vertices with the maximal Merrifield-Simmons index among all graphs ...

We investigate the zero-forcing number for triangle-free graphs. We improve upon the trivial bound, δ ≤ Z(G) where δ is the minimum degree, in the triangle-free case. In particular, we show that 2δ− 2 ≤ Z(G) for graphs with girth of at least 5, and this can be further improved when G has a small cut set. Lastly, we make a conjecture that the lower bound for Z(G) increases as a function of the g...

Nilpotent and idempotent adjacency operator methods are applied to the study of random geometric graphs in a discretized, d-dimensional unit cube [0, 1]. Cycles are enumerated, sizes of maximal connected components are computed, and closed formulas are obtained for graph circumference and girth. Expected numbers of k-cycles, expected sizes of maximal components, and expected circumference and g...

Let G be a graph and let c : V (G) → ({1,...,5} 2 ) be an assignment of 2-element subsets of the set {1, . . . , 5} to the vertices of G such that for every edge vw, the sets c(v) and c(w) are disjoint. We call such an assignment a (5, 2)-coloring. A graph is (5,2)-colorable if and only if it has a homomorphism to the Petersen graph. The odd-girth of a graph G is the length of the shortest odd ...

We study the problem of recognizing graph powers and computing roots of graphs. We provide a polynomial time recognition algorithm for r-th powers of graphs of girth at least 2r + 3, thus improving a bound conjectured by Farzad et al. (STACS 2009). Our algorithm also finds all r-th roots of a given graph that have girth at least 2r+3 and no degree one vertices, which is a step towards a recent ...

We show a simple deterministic linear time algorithm for computing the minimum spanning tree of graphs on n vertices with girth at least b(t, lg n), where t ≥ 1 is a constant and b(x, y) is a variant of the inverse of the fast-growing Ackermann function. We also prove: (i) A deterministic linear time algorithm for general graphs follows from an algorithm for graphs with girth at least g for any...

Jeager et al introduced a concept of group connectivity as an generalization of nowhere zero flows and its dual concept group coloring, and conjectured that every 5-edge connected graph is Z3-connected. For planar graphs, this is equivalent to that every planar graph with girth at least 5 must have group chromatic number at most 3. In this paper we show that if G is a plane graph with girth at ...

A connected graph of girth m ≥ 3 is called a polygonal graph if it contains a set of m-gons such that every path of length two is contained in a unique element of the set. In this paper we investigate polygonal graphs of girth 6 or more having automorphism groups which are transitive on the vertices and such that the vertex stabilizers are 3-homogeneous on adjacent vertices. We previously showe...

The odd-girth of a graph is the length of a shortest odd circuit. A conjecture by Pavol Hell about circular coloring is solved in this article by showing that there is a function f( ) for each : 0 < < 1 such that, if the odd-girth of a planar graph G is at least f( ), then G is (2 + )-colorable. Note that the function f( ) is independent of the graph G and → 0 if and only if f( )→∞. A key lemma...

A star-factor of a graph G is a spanning subgraph of G such that each component of which is a star. Recently, Hartnell and Rall studied a family U of graphs satisfying the property that every star-factor of a member graph has the same number of edges. They determined the family U when the girth is at least five. In this paper, we investigate the family of graphs with girth three and determine a...