A graph is integral if the spectrum (of its adjacency matrix) consists entirely of integers. The problem of determining all non-regular bipartite integral graphs with maximum degree four which do not have ±1 as eigenvalues was posed in K.T. Balińska, S.K. Simić, K.T. Zwierzyński: Which nonregular bipartite integral graphs with maximum degree four do not have ±1 as eigenvalues? Discrete Math., 2...

Let G be a non-trivial group, S ⊆ G \ {1} and S = S−1 := {s−1 | s ∈ S}. The Cayley graph of G denoted by Γ(S : G) is a graph with vertex set G and two vertices a and b are adjacent if ab−1 ∈ S. A graph is called integral, if its adjacency eigenvalues are integers. In this paper we determine all connected cubic integral Cayley graphs. We also introduce some infinite families of connected integra...

A mixed graph is called second kind hermitian integral (or HS-integral) if the eigenvalues of its Hermitian-adjacency matrix are integers. Eisenstein (0, 1)-adjacency Let ? be an abelian group. We characterize set S for which a Cayley Cay(?,S) HS-integral. also show that and only it

The purpose of this paper is to present some coupled fixed point results on a metric space endowed with two $b$-metrics. We shall apply a fixed point theorem for an appropriate operator on the Cartesian product of the given spaces endowed with directed graphs. Data dependence, well-posedness and Ulam-Hyers stability are also studied. The results obtained here will be applied to prove the existe...

We propose new families of graphs which exhibit quantum perfect state transfer. Our constructions are based on the join operator on graphs, its circulant generalizations, and the Cartesian product of graphs. We build upon the results of Bašić and Petković (Applied Mathematics Letters 22(10):1609-1615, 2009) and construct new integral circulants and regular graphs with perfect state transfer. Mo...

Let D be the diameter of a graph G and let λ1 be the largest eigenvalue of its (0,1)adjacency matrix. We give a proof of the fact that there are exactly 69 non-trivial connected graphs with (D + 1)λ1 9. These 69 graphs all have up to 10 vertices and were recently found to be suitable models for small multiprocessor interconnection networks. We also examine the suitability of integral graphs to ...

In this paper we study the maximum two-flow problem in vertexand edge-capacitated undirected STI-planar graphs, that is, planar graphs where the vertices of each terminal pair arc on the same face. For such graphs we provide an O(n) algorithm for finding a minimum two-cut and an O(n log n) algorithm for determining a maximum two-flow and show that the value of a maximum two-flow equals the valu...

The energy of a graph is the sum of the moduli of the eigenvalues of its adjacency matrix. We study the energy of integral circulant graphs, also called gcd graphs, which can be characterized by their vertex count n and a set D of divisors of n in such a way that they have vertex set Zn and edge set {{a, b} : a, b ∈ Zn, gcd(a− b, n) ∈ D}. Using tools from convex optimization, we study the maxim...

Throughout this paper a graph G is assumed to be simple, i.e. a finite undirected graph without loops or multiple edges. Therefore, the characteristic polynomial of (the adjacency matrix of) G, denoted by PG(λ), has only real zeroes and this family of eigenvalues (the spectrum of G) will be represented as (λ1, λ2, . . . , λn), where λ1 ≥ λ2 ≥ · · · ≥ λn or in the form μ1 1 , μ2 2 , . . . , μm m...

We consider in this note the maximum integral two-commodity flow problem in augmented planar graphs, that is with both sourceesink edges added. We provide an O(n&) simple algorithm for finding a maximum integral two-commodity flow in augmented planar graphs.