Edge Regular Graph Products

A regular nonempty graph $\Gamma$ is called edge regular, whenever there exists a nonegative integer $\lambda_{\Gamma}$, such that any two adjacent vertices of $\Gamma$ have precisely $\lambda_{\Gamma}$ common neighbours. An edge regular graph $\Gamma$ with at least one pair of vertices at distance 2 is called amply regular, whenever there exists a nonegative integer $\mu_{\Gamma}$, such that any two vertices at distance 2 have precisely $\mu_{\Gamma}$ common neighbours. In this paper we classify edge regular graphs, which can be obtained as a strong product, or a lexicographic product, or a deleted lexicographic product, or a co-normal product of two graphs. As a corollary we determine which of these graphs are amply regular.

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On the b-chromatic number of some graph products

Studia Scientiarum Mathematicarum Hungarica ◽

10.1556/sscmath.49.2012.2.1194 ◽

2012 ◽

Vol 49 (2) ◽

pp. 156-169 ◽

Cited By ~ 4

Author(s):

Marko Jakovac ◽

Iztok Peterin

Keyword(s):

Chromatic Number ◽

Upper And Lower Bounds ◽

Vertex Coloring ◽

Graph Products ◽

Lexicographic Product ◽

Arbitrary Graph ◽

Strong Product ◽

Color Class ◽

Complete Bipartite Graphs ◽

Proper Vertex

A b-coloring is a proper vertex coloring of a graph such that each color class contains a vertex that has a neighbor in all other color classes and the b-chromatic number is the largest integer φ(G) for which a graph has a b-coloring with φ(G) colors. We determine some upper and lower bounds for the b-chromatic number of the strong product G ⊠ H, the lexicographic product G[H] and the direct product G × H and give some exact values for products of paths, cycles, stars, and complete bipartite graphs. We also show that the b-chromatic number of Pn ⊠ H, Cn ⊠ H, Pn[H], Cn[H], and Km,n[H] can be determined for an arbitrary graph H, when integers m and n are large enough.

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On the Laplacian spectra of product graphs

Applicable Analysis and Discrete Mathematics ◽

10.2298/aadm150218006b ◽

2015 ◽

Vol 9 (1) ◽

pp. 39-58 ◽

Cited By ~ 12

Author(s):

S. Barik ◽

R.B. Bapat ◽

S. Pati

Keyword(s):

Cartesian Product ◽

Complete Characterization ◽

Graph Products ◽

Lexicographic Product ◽

Laplacian Spectrum ◽

Strong Product ◽

Laplacian Spectra ◽

Product Graphs ◽

Characteristic Sets

Graph products and their structural properties have been studied extensively by many researchers. We investigate the Laplacian eigenvalues and eigenvectors of the product graphs for the four standard products, namely, the Cartesian product, the direct product, the strong product and the lexicographic product. A complete characterization of Laplacian spectrum of the Cartesian product of two graphs has been done by Merris. We give an explicit complete characterization of the Laplacian spectrum of the lexicographic product of two graphs using the Laplacian spectra of the factors. For the other two products, we describe the complete spectrum of the product graphs in some particular cases. We supply some new results relating to the algebraic connectivity of the product graphs. We describe the characteristic sets for the Cartesian product and for the lexicographic product of two graphs. As an application we construct new classes of Laplacian integral graphs.

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Graph Products and New Solutions to Oberwolfach Problems

The Electronic Journal of Combinatorics ◽

10.37236/539 ◽

2011 ◽

Vol 18 (1) ◽

Cited By ~ 8

Author(s):

Gloria Rinaldi ◽

Tommaso Traetta

Keyword(s):

Regular Graph ◽

Regular Graphs ◽

Graph Products ◽

Simple Graphs ◽

Oberwolfach Problem

We introduce the circle product, a method to construct simple graphs starting from known ones. The circle product can be applied in many different situations and when applied to regular graphs and to their decompositions, a new regular graph is obtained together with a new decomposition. In this paper we show how it can be used to construct infinitely many new solutions to the Oberwolfach problem, in both the classic and the equipartite case.

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Dominating Sequences in Grid-Like and Toroidal Graphs

The Electronic Journal of Combinatorics ◽

10.37236/6269 ◽

2016 ◽

Vol 23 (4) ◽

Cited By ~ 3

Author(s):

Boštjan Brešar ◽

Csilla Bujtás ◽

Tanja Gologranc ◽

Sandi Klavžar ◽

Gašper Košmrlj ◽

...

Keyword(s):

Lower Bound ◽

Domination Number ◽

Graph Products ◽

Lexicographic Product ◽

Strong Product ◽

Toroidal Graphs ◽

Standard Graph ◽

Domination Numbers

A longest sequence $S$ of distinct vertices of a graph $G$ such that each vertex of $S$ dominates some vertex that is not dominated by its preceding vertices, is called a Grundy dominating sequence; the length of $S$ is the Grundy domination number of $G$. In this paper we study the Grundy domination number in the four standard graph products: the Cartesian, the lexicographic, the direct, and the strong product. For each of the products we present a lower bound for the Grundy domination number which turns out to be exact for the lexicographic product and is conjectured to be exact for the strong product. In most of the cases exact Grundy domination numbers are determined for products of paths and/or cycles.

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On Self-Centeredness of Product of Graphs

International Journal of Combinatorics ◽

10.1155/2016/2508156 ◽

2016 ◽

Vol 2016 ◽

pp. 1-4 ◽

Cited By ~ 1

Author(s):

Priyanka Singh ◽

Pratima Panigrahi

Keyword(s):

Finite Number ◽

Sufficient Conditions ◽

Necessary And Sufficient Conditions ◽

Lexicographic Product ◽

Normal Product ◽

Strong Product ◽

Necessary And Sufficient ◽

Product Of Graphs

A graph G is said to be a self-centered graph if the eccentricity of every vertex of the graph is the same. In other words, a graph is a self-centered graph if radius and diameter of the graph are equal. In this paper, self-centeredness of strong product, co-normal product, and lexicographic product of graphs is studied in detail. The necessary and sufficient conditions for these products of graphs to be a self-centered graph are also discussed. The distance between any two vertices in the co-normal product of a finite number of graphs is also computed analytically.

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The $A_{\alpha}$- spectrum of graph product

Electronic Journal of Linear Algebra ◽

10.13001/1081-3810.3857 ◽

2019 ◽

Vol 35 ◽

pp. 473-481 ◽

Cited By ~ 2

Author(s):

Shuchao Li ◽

Shujing Wang

Keyword(s):

Regular Graph ◽

Cartesian Product ◽

Complete Characterization ◽

Lexicographic Product ◽

Arbitrary Graph ◽

Strong Product ◽

Alpha Spectrum ◽

Vertex Degrees ◽

Arbitrary Graphs

Let $A(G)$ and $D(G)$ denote the adjacency matrix and the diagonal matrix of vertex degrees of $G$, respectively. Define $$ A_{\alpha}(G)=\alpha D(G)+(1-\alpha)A(G) $$ for any real $\alpha\in [0,1]$. The collection of eigenvalues of $A_{\alpha}(G)$ together with multiplicities is called the $A_{\alpha}$-\emph{spectrum} of $G$. Let $G\square H$, $G[H]$, $G\times H$ and $G\oplus H$ be the Cartesian product, lexicographic product, directed product and strong product of graphs $G$ and $H$, respectively. In this paper, a complete characterization of the $A_{\alpha}$-spectrum of $G\square H$ for arbitrary graphs $G$ and $H$, and $G[H]$ for arbitrary graph $G$ and regular graph $H$ is given. Furthermore, $A_{\alpha}$-spectrum of the generalized lexicographic product $G[H_1,H_2,\ldots,H_n]$ for $n$-vertex graph $G$ and regular graphs $H_i$'s is considered. At last, the spectral radii of $A_{\alpha}(G\times H)$ and $A_{\alpha}(G\oplus H)$ for arbitrary graph $G$ and regular graph $H$ are given.

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Cycle partitions of regular graphs

Combinatorics Probability Computing ◽

10.1017/s0963548320000553 ◽

2020 ◽

pp. 1-24

Author(s):

Vytautas Gruslys ◽

Shoham Letzter

Keyword(s):

Regular Graph ◽

Bipartite Graphs ◽

Disjoint Paths ◽

Regular Graphs ◽

Simple Construction ◽

Vertex Set ◽

Almost All ◽

Vertex Disjoint

Abstract Magnant and Martin conjectured that the vertex set of any d-regular graph G on n vertices can be partitioned into $n / (d+1)$ paths (there exists a simple construction showing that this bound would be best possible). We prove this conjecture when $d = \Omega(n)$ , improving a result of Han, who showed that in this range almost all vertices of G can be covered by $n / (d+1) + 1$ vertex-disjoint paths. In fact our proof gives a partition of V(G) into cycles. We also show that, if $d = \Omega(n)$ and G is bipartite, then V(G) can be partitioned into n/(2d) paths (this bound is tight for bipartite graphs).

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The maximum number of vertices of primitive regular graphs of orders 2, 3, 4 with exponent 2

PRIKLADNAYa DISKRETNAYa MATEMATIKA ◽

10.17223/20710410/52/6 ◽

2021 ◽

pp. 97-104

Author(s):

M. B. Abrosimov ◽

◽

S. V. Kostin ◽

I. V. Los ◽

◽

...

Keyword(s):

Regular Graph ◽

Regular Graphs ◽

Minimal Regular Graphs of Girths Eight and Twelve

Canadian Journal of Mathematics ◽

10.4153/cjm-1966-109-8 ◽

1966 ◽

Vol 18 ◽

pp. 1091-1094 ◽

Cited By ~ 104

Author(s):

Clark T. Benson

Keyword(s):

Group Theory ◽

Lower Bound ◽

Regular Graph ◽

Regular Graphs ◽

Image Position

In (3) Tutte showed that the order of a regular graph of degree d and even girth g > 4 is greater than or equal toHere the girth of a graph is the length of the shortest circuit. It was shown in (2) that this lower bound cannot be attained for regular graphs of degree > 2 for g ≠ 6, 8, or 12. When this lower bound is attained, the graph is called minimal. In a group-theoretic setting a similar situation arose and it was noticed by Gleason that minimal regular graphs of girth 12 could be constructed from certain groups. Here we construct these graphs making only incidental use of group theory. Also we give what is believed to be an easier construction of minimal regular graphs of girth 8 than is given in (2). These results are contained in the following two theorems.

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On the Number of Spanning Trees in Random Regular Graphs

The Electronic Journal of Combinatorics ◽

10.37236/3752 ◽

2014 ◽

Vol 21 (1) ◽

Author(s):

Catherine Greenhill ◽

Matthew Kwan ◽

David Wind

Keyword(s):

Asymptotic Distribution ◽

Regular Graph ◽

Spanning Trees ◽

Regular Graphs ◽

Cubic Graph ◽

Expected Number ◽

Fixed Integer ◽

Numerical Evidence ◽

Number Of Spanning Trees

Let $d\geq 3$ be a fixed integer. We give an asympotic formula for the expected number of spanning trees in a uniformly random $d$-regular graph with $n$ vertices. (The asymptotics are as $n\to\infty$, restricted to even $n$ if $d$ is odd.) We also obtain the asymptotic distribution of the number of spanning trees in a uniformly random cubic graph, and conjecture that the corresponding result holds for arbitrary (fixed) $d$. Numerical evidence is presented which supports our conjecture.

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