Nowadays, the problem of finding families graphs for which one may ensure existence a vertex-labeling and/or an edge-labeling based on certain class integers, constitutes challenge researchers in both number and graph theory. In this paper, we focus those vertex-labelings whose induced multiplicative assigns hyper totient numbers to edges graph. way, introduce characterize notions re...

The problem of distinguishing, in terms of graph topology, digraphs with real and partially non-real Laplacian spectra is important for applications. Motivated by the question posed in [R. Agaev, P. Chebotarev, Which digraphs with rings structure are essentially cyclic?, Adv. in Appl. Math. 45 (2010), 232–251], in this paper we completely list the Laplacian eigenvalues of some digraphs obtained...

A generalized vertex join of a graph is obtained by joining an arbitrary multiset of its vertices to a new vertex. We present a low-order polynomial time algorithm for finding the chromatic polynomials of generalized vertex joins of trees, and by duality we find the flow polynomials of arbitrary outerplanar graphs. We also present closed formulas for the chromatic and flow polynomials of vertex...

For two given graphs G1 and G2, the Ramsey number R(G1,G2) is the smallest positive integer n such that for any graph G of order n, either G contains G1 or the complement of G contains G2. Let Sn denote a star of order n and Wm a wheel of order m + 1. In this paper, we show that R(Sn,W8) = 2n + 2 for n ≥ 6 and n ≡ 0 (mod 2). c © 2007 Elsevier Ltd. All rights reserved.

Abstract A fully connected vertex w in a simple graph G of order N is to all the other ? 1 vertices. Upon denoting by L Laplacian matrix graph, we prove that continuous-time quantum walk (CTQW)—with Hamiltonian H = ?L —of walker initially localized at | ? does not depend on . We also for any Grover-like CTQW—with + ? ? ?? |—the probability amplitude marked vertices The result hold CTQW with ?A ...

In this article, we characterize the graphs G that are the retracts of Cartesian products of chordal graphs. We show that they are exactly the weakly modular graphs that do not contain K2,3, the 4-wheel minus one spoke W − 4 , and the k-wheels Wk (for k ≥ 4) as induced subgraphs. We also show that these graphs G are exactly the cage-amalgamation graphs as introduced by Brešar and Tepeh Horvat (...

Let G be a graph with vertex set V(G) and edge set E(G), and let A be an abelian group. A labeling f : V(G) ....... A induces a edge labeling r : E{G) ....... A defined by r(xy) = f(x) + f(y) for each xy E E. For each i E A, let vJ(i) card{v E V(G) : f(v) i} and eJ(i) card{e E E(G) : r(e) i}. Let c(J) {leJ(i) eJ(j)1 : = = (i,j) E A x A}. A labeling f of a graph G is said to be A-friendly if IVJ...

By the sorting method of vertices, the equitable chromatic number and the equitable chromatic threshold of the Cartesian products of wheels with bipartite graphs are obtained. Key–Words: Cartesian product, Equitable coloring, Equitable chromatic number, Equitable chromatic threshold

The subgraph homeomorphism problem, SHP(H), has been shown to be polynomial-time solvable for any fixed pattern graph H, but practical algorithms have been developed only for a few specific pattern graphs. Among these are the wheels with four, five, and six spokes. This paper examines the subgraph homeomorphism problem where the pattern graph is a wheel with seven spokes, and gives a result tha...

In this work, we study the signed Roman domination number of the join of graphs. Specially, we determine it for the join of cycles, wheels, fans, and friendship graphs.