Circulant Graphs Research Articles

The number of spanning trees of the bipartite complement of a semiregular bipartite graph

For a bipartite graph $G$ on parts of cardinality $m$ and $n$, the bipartite complement of $G$ is defined as the graph obtained from the complete bipartite graph $K_{m,n}$ by removing the edges of an isomorph of $G$. In this paper, based on the matrix-tree theorem and the Schur complement technique, we give a determinant expression for the number of spanning trees of the bipartite complement of a semiregular bipartite graph. As by-products, we show that the corresponding result can generalize some previous results on the problem of enumerating spanning trees and obtain an explicit formula for the number of spanning trees of the bipartite complement of the subdivison of a regular circulant graph.

Optimizing network insights: AI-Driven approaches to circulant graph based on Laplacian spectra

The study of Laplacian and signless Laplacian spectra extends across various fields, including theoretical chemistry, computer science, electrical networks, and complex networks, providing critical insights into the structures of real-world networks and enabling the prediction of their structural properties. A key aspect of this study is the spectrum-based analysis of circulant graphs. Through these analyses, important network measures such as mean-first passage time, average path length, spanning trees, and spectral radius are derived. This research enhances our understanding of the relationship between graph spectra and network characteristics, offering a comprehensive perspective on complex networks. Consequently, it supports the ability to make predictions and conduct analyses across a wide range of scientific disciplines.

Fractional revival on Cayley graphs over abelian groups

In this paper, we investigate the existence of fractional revival on Cayley graphs over finite abelian groups. We give a necessary and sufficient condition for Cayley graphs over finite abelian groups to have fractional revival. As applications, the existence of fractional revival on circulant graphs and cubelike graphs are characterized.

Maximal diameter of integral circulant graphs

Integral circulant graphs are proposed as models for quantum spin networks enabling perfect state transfer. Understanding the potential information transfer between nodes in such networks involves calculating the maximal graph diameter. The integral circulant graph ICGn(D) has vertex set Zn={0,1,2,…,n−1}, with vertices a and b adjacent if gcd⁡(a−b,n)∈D, where D⊆{d:d|n,1≤d<n}. Building on the upper bound 2|D|+1 for the diameter provided by Saxena, Severini, and Shparlinski, we prove that the maximal diameter of ICGn(D) for a given order n with prime factorization p1α1⋯pkαk is r(n) or r(n)+1, where r(n)=k+|{i|αi>1,1≤i≤k}|. We show that a divisor set D with |D|≤k achieves this bound. We calculate the maximal diameter for graphs of order n and divisor set cardinality t≤k, identifying all extremal graphs and improving the previous upper bound.

On the structure of Laplacian characteristic polynomial of circulant graphs

On the structure of Laplacian characteristic polynomial of circulant graphs

Computing Edge Irregularity Strength of Star and Banana Trees Using Algorithmic Approach

After the Chartrand definition of graph labeling, since 1988 many graph families have been labeled through mathematical techniques. A basic approach in those labelings was to find a pattern among the labels and then prove them using sequences and series formulae. In 2018, Asim applied computer-based algorithms to overcome this limitation and label such families where mathematical solutions were either not available or the solution was not optimum. Asim et al. in 2018 introduced the algorithmic solution in the area of edge irregular labeling for computing a better upper-bound of the complete graph \(es(K_n)\) and a tight upper-bound for the complete \(m\)-ary tree \({es(T}_{m,h})\) using computer-based experiments. Later on, more problems like complete bipartite and circulant graphs were solved using the same technique. Algorithmic solutions opened a new horizon for researchers to customize these algorithms for other types of labeling and for more complex graphs. In this article, to compute edge irregular \(k\)-labeling of star \(S_{m,n}\) and banana tree \({BT}_{m,n}\), new algorithms are designed, and results are obtained by executing them on computers. To validate the results of computer-based experiments with mathematical theorems, inductive reasoning is adopted. Tabulated results are analyzed using the law of double inequality and it is concluded that both families of trees observe the property of edge irregularity strength and are tight for \(\left\lceil \frac{|V|}{2} \right\rceil\)-labeling.

Regular sets in circulant graphs with degree 3

Let Γ=(V,E) be a graph and a,b nonnegative integers. An (a,b)-regular set in Γ is a nonempty proper subset D of V such that every vertex in D has exactly a neighbours in D and every vertex in V∖D has exactly b neighbours in D. In this paper, we obtain a necessary and sufficient condition for the existence of (a,b)-regular sets in connected circulant graphs with degree 3, for all possible values a,b.

Further studies on circulant completion of graphs

A circulant graph C(n,S) is a graph having its adjacency matrix as a circulant matrix. It can also be intrepreted as a graph with vertices v0,v1,...,vn-1 that are in one to one correspondence with the members of Zn and with edge set {vivj:i-j ∈ S}, where S known as the connection set or symbol, is a subset of non-identity members of Zn that is closed under inverses. This work extends the study of circulant completion and general formulae for calculating circulant completion number in two different perspectives, one in terms of circulant span and the other in terms of adjacency matrix.

New qubit codes from multidimensional circulant graphs

Two new qubit stabilizer codes with parameters 〚77,0,19〛2 and 〚90,0,22〛2 are constructed for the first time by employing additive symplectic self-dual F4 codes from multidimensional circulant (MDC) graphs. We completely classify MDC graph codes for lengths 4≤n≤40 and show that many optimal 〚ℓ,0,d〛 qubit codes can be obtained from the MDC construction. Moreover, we prove that adjacency matrices of MDC graphs have nested block circulant structure and determine isomorphism properties of MDC graphs.

Metric Dimension of Circulant Graphs with 5 Consecutive Generators

The problem of finding the metric dimension of circulant graphs with t generators 1,2,…,t (and their inverses) has been extensively studied. The problem is solved for t=2,3,4, and some exact values and bounds are known also for t=5. We solve all the open cases for t=5.

Optimal Layout of (Kp − Cp)n into Wheel-Like Networks

The problem of finding the induced subgraph with the maximum number of edges among all subsets of a fixed cardinality is known as the maximum subgraph problem (MSP). The lexicographic order has been proven optimal for Cartesian products of specific regular graphs [B. Sergei, B. Pavle and K. Nikola, New infinite family of regular edge isoperimetric graphs, Theoretical Computer Science 721 (2018) 42–53]. This paper is dedicated to the precise calculation of edge values within the resulting subgraphs, enhancing our comprehension of their structural properties and implications. In this paper, we focus on solving the MSP of [Formula: see text] for any [Formula: see text] and for odd [Formula: see text], [Formula: see text] and [Formula: see text], where [Formula: see text] is obtained by the Cartesian product of a complete graph with [Formula: see text] vertices with a removal of a cycle on [Formula: see text] vertices. It has been observed that the graph obtained by deleting a cycle of length [Formula: see text] from a complete graph [Formula: see text] is isomorphic to a circulant graph [Formula: see text], while the particular case [Formula: see text] has been previously studied in [A. Syeda and M. Rajesh, MinLA of [Formula: see text] and its optimal layout into certain trees, The Journal of Supercomputing (2023), https://doi.org/10.1007/s11227-023-05140-3]. Our work extends the understanding of the MSP to a broader class of graphs with similar properties and applications.

So's conjecture for integral circulant graphs of 4 types

In [Discrete Mathematics 306 (2005) 153-158], So proposed a conjecture saying that integral circulant graphs with different connection sets have different spectra. This conjecture is still open. We prove that this conjecture holds for integral circulant graphs whose orders have prime factorization of 4 types.

A note on bounds for the broadcast domination number of graphs

A dominating broadcast of a graph G is a function f:V(G)→{0,1,2,…,diam(G)} such that f(v)⩽e(v) for all v∈V(G), where e(v) is the eccentricity of v, and for every u∈V(G), there exists a vertex v with f(v)⩾1 and d(u,v)⩽f(v). The cost of f is ∑v∈V(G)f(v). The minimum cost over all the dominating broadcasts of G is called the broadcast domination number γb(G) of G. In this paper, we give new upper and lower bounds for γb(G). We show that both the bounds are tight. Also, we improve the upper bound for a subclass of regular graphs. Further, we explore the broadcast domination numbers of generalized Petersen graphs and a subclass of circulant graphs.

On the Structure of Laplacian Characteristic Polynomial of Circulant Graphs

On the Structure of Laplacian Characteristic Polynomial of Circulant Graphs

Routing in circulant graphs based on a virtual coordinate system

Routing in circulant graphs based on a virtual coordinate system

On the WL-dimension of circulant graphs of prime power order

On the WL-dimension of circulant graphs of prime power order

Distance antimagic labeling of circulant graphs

A distance antimagic labeling of graph $ G = (V, E) $ of order $ n $ is a bijection $ f:V(G)\rightarrow \{1, 2, \ldots, n\} $ with the property that any two distinct vertices $ x $ and $ y $ satisfy $ \omega(x)\ne\omega(y) $, where $ \omega(x) $ denotes the open neighborhood sum $ \sum_{a\in N(x)}f(a) $ of a vertex $ x $. In 2013, Kamatchi and Arumugam conjectured that a graph admits a distance antimagic labeling if and only if it contains no two vertices with the same open neighborhood. A circulant graph $ C(n; S) $ is a Cayley graph with order $ n $ and generating set $ S $, whose adjacency matrix is circulant. This paper provides partial evidence for the conjecture above by presenting distance antimagic labeling for some circulant graphs. In particular, we completely characterized distance antimagic circulant graphs with one generator and distance antimagic circulant graphs $ C(n; \{1, k\}) $ with odd $ n $.

On (r,c)-constant, planar and circulant graphs

On (r,c)-constant, planar and circulant graphs

Algebraic invariants of edge ideals of some circulant graphs

&lt;abstract&gt;&lt;p&gt;Let $ S $ be a polynomial ring over a field and $ I $ be an edge ideal associated with some classes of circulant graphs. We discussed the algebraic invariants, namely, regularity, projective dimension, depth, and the Stanley depth of $ S/I. $&lt;/p&gt;&lt;/abstract&gt;

On Dispersability of Some Circulant Graphs

The matching book thickness of a graph is the least number of pages in a book embedding such that each page is a matching. A graph is dispersable if its matching book thickness equals its maximum degree. Minimum page matching book embeddings are given for bipartite and for most non-bipartite circulants contained in the (Harary) cube of a cycle and for various higher-powers.