A family of graphs possesses the common gcd property if the greatest common divisor of the degree sequence of each graph in the family is the same. In particular, any family of trees has the common gcd property. Let F = {H1, . . . ,Hr} be a family of graphs having the common gcd property, and let d be the common gcd. It is proved that there exists a constant N = N(F ) such that for every n > N ...

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...

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In this paper, two outwardly different graphs, namely, the zero-divisor graph [Formula: see text] and comaximal of ring all real-valued continuous functions having countable range, defined on any zero-dimensional space text], are investigated. It is observed that these graphs exhibit resemblance, so far as diameters, girths, connectedness, triangulatedness or hypertriangulatedness concerned. Ho...

Connected cubic (m,n)-metacircu1ant graphs, other than the Petersen , have been to be hamiltonian for m m divisible by 4 and m = 2. In this paper we two sufficient conditions tor connected cubic-metac1rcu1ant with m even, than 2 and not divisible by 4 to be hamiltonian. As corQ1laries, we show that every connected cubic-metacircu1ant other than the Petersen ,has a Hamilton if anyone of the cond...

We elaborate upon a bijection discovered by Cools, Draisma, Payne, and Robeva in [CDPR12] between the set of rectangular standard Young tableaux and the set of equivalence classes of chip configurations on certain metric graphs under the relation of linear equivalence. We present an explicit formula for computing the v0-reduced divisors (representatives of the equivalence classes) associated to...

Baker and Norine initiated the study of graph divisors as a graph-theoretic analogue Riemann-Roch theory for Riemann surfaces. One key concepts divisor is rank on graph. The importance well illustrated by Baker's Specialization lemma, stating that dimension linear system can only go up under specialization from curves to graphs, leading fruitful interaction between graphs curves. Due its decisi...

We study further CCZ-equivalence of (n,m)-functions. We prove that for Boolean functions (that is, for m = 1), CCZ-equivalence coincides with EA-equivalence. On the contrary, we show that for (n,m)functions, CCZ-equivalence is strictly more general than EAequivalence when n ≥ 5 and m is greater or equal to the smallest positive divisor of n different from 1. Our result on Boolean functions allo...

We denote by r'( G I , G2 ) the ramsey number of two graphs and G2 • If Tp+l is a tree of order p + 1 which is not a star, and if jJ is not a divisor of the positive integer q 1, then we shall show that r(Tp+b ](l,q) ::; p + q 1, and we shall describe some trees and stars for which equality holds. Furthermore. we detf'rmine the ramsey numbers r(T;+l' Tq*+l) for p, q ~ 4, where T~ denotes a tree...