Some Properties of Regular Line Graphs

Regular Line ◽

In this paper, the concept of regular line graph has been introduced. The maximum number of vertices with different degrees in the regular line graphs has also been studied. Further, the necessary and sufficient condition for regular line graph to be bipartite graph have also been proved. Key words: Line Graphs, Regular graphs, Connected graphs, Bipartite Graphs.

The competition number of a generalized line graph is at most two

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.575 ◽

2012 ◽

Vol Vol. 14 no. 2 (Graph Theory) ◽

Author(s):

Boram Park ◽

Yoshio Sano

Keyword(s):

Graph Theory ◽

Necessary Conditions ◽

Line Graph ◽

Sufficient Conditions ◽

Line Graphs ◽

Sufficient Condition ◽

International Audience ◽

Competition Number

Graph Theory International audience In 1982, Opsut showed that the competition number of a line graph is at most two and gave a necessary and sufficient condition for the competition number of a line graph being one. In this paper, we generalize this result to the competition numbers of generalized line graphs, that is, we show that the competition number of a generalized line graph is at most two, and give necessary conditions and sufficient conditions for the competition number of a generalized line graph being one.

Bipartite graphs with close domination and k-domination numbers

Open Mathematics ◽

10.1515/math-2020-0047 ◽

2020 ◽

Vol 18 (1) ◽

pp. 873-885

Author(s):

Gülnaz Boruzanlı Ekinci ◽

Csilla Bujtás

Keyword(s):

Positive Integer ◽

Dominating Set ◽

Domination Number ◽

Bipartite Graphs ◽

Sufficient Condition ◽

Minimum Cardinality ◽

Vertex Set ◽

The Given

Abstract Let k be a positive integer and let G be a graph with vertex set V(G) . A subset D\subseteq V(G) is a k -dominating set if every vertex outside D is adjacent to at least k vertices in D . The k -domination number {\gamma }_{k}(G) is the minimum cardinality of a k -dominating set in G . For any graph G , we know that {\gamma }_{k}(G)\ge \gamma (G)+k-2 where \text{Δ}(G)\ge k\ge 2 and this bound is sharp for every k\ge 2 . In this paper, we characterize bipartite graphs satisfying the equality for k\ge 3 and present a necessary and sufficient condition for a bipartite graph to satisfy the equality hereditarily when k=3 . We also prove that the problem of deciding whether a graph satisfies the given equality is NP-hard in general.

Unitary homogeneous bi-Cayley graphs over finite commutative rings

Journal of Algebra and Its Applications ◽

10.1142/s021949882050173x ◽

2019 ◽

Vol 19 (09) ◽

pp. 2050173

Author(s):

Xiaogang Liu ◽

Chengxin Yan

Keyword(s):

Commutative Ring ◽

Cayley Graph ◽

Line Graph ◽

Cayley Graphs ◽

Commutative Rings ◽

Sufficient Condition ◽

Finite Commutative Ring ◽

Finite Commutative Rings

Let [Formula: see text] denote the unitary homogeneous bi-Cayley graph over a finite commutative ring [Formula: see text]. In this paper, we determine the energy of [Formula: see text] and that of its complement and line graph, and characterize when such graphs are hyperenergetic. We also give a necessary and sufficient condition for [Formula: see text] (respectively, the complement of [Formula: see text], the line graph of [Formula: see text]) to be Ramanujan.

Families of Integral Cographs within a Triangular Array

Special Matrices ◽

10.1515/spma-2020-0116 ◽

2020 ◽

Vol 8 (1) ◽

pp. 257-273

Author(s):

Hsin-Yun Ching ◽

Rigoberto Flórez ◽

Antara Mukherjee

Keyword(s):

Triangular Array ◽

Sufficient Condition ◽

Complete Graphs ◽

Regular Graphs ◽

Empty Graph ◽

Complete Product ◽

AbstractThe determinant Hosoya triangle, is a triangular array where the entries are the determinants of two-by-two Fibonacci matrices. The determinant Hosoya triangle mod 2 gives rise to three infinite families of graphs, that are formed by complete product (join) of (the union of) two complete graphs with an empty graph. We give a necessary and sufficient condition for a graph from these families to be integral.Some features of these graphs are: they are integral cographs, all graphs have at most five distinct eigenvalues, all graphs are either d-regular graphs with d =2, 4, 6, . . . or almost-regular graphs, and some of them are Laplacian integral. Finally we extend some of these results to the Hosoya triangle.

Even Cycles and Even 2-Factors in the Line Graph of a Simple Graph

The Electronic Journal of Combinatorics ◽

10.37236/5660 ◽

2017 ◽

Vol 24 (4) ◽

Cited By ~ 1

Author(s):

Arrigo Bonisoli ◽

Simona Bonvicini

Keyword(s):

Perfect Matching ◽

Line Graph ◽

Simple Graph ◽

Connected Components ◽

Regular Graphs ◽

Cubic Graph ◽

Cycle Decomposition ◽

Odd Degree ◽

Let $G$ be a connected graph with an even number of edges. We show that if the subgraph of $G$ induced by the vertices of odd degree has a perfect matching, then the line graph of $G$ has a $2$-factor whose connected components are cycles of even length (an even $2$-factor). For a cubic graph $G$, we also give a necessary and sufficient condition so that the corresponding line graph $L(G)$ has an even cycle decomposition of index $3$, i.e., the edge-set of $L(G)$ can be partitioned into three $2$-regular subgraphs whose connected components are cycles of even length. The more general problem of the existence of even cycle decompositions of index $m$ in $2d$-regular graphs is also addressed.

The degree sequences of a graph with restrictions

Indonesian Journal of Combinatorics ◽

10.19184/ijc.2021.5.2.2 ◽

2021 ◽

Vol 5 (2) ◽

pp. 68

Author(s):

Rikio Ichishima ◽

Francesc A. Muntaner-Batle ◽

Miquel Rius-Font ◽

Yukio Takahashi

Keyword(s):

Bipartite Graph ◽

Sufficient Condition ◽

Degree Sequences ◽

Finite Sequences ◽

Two finite sequences s1 and s2 of nonnegative integers are called bigraphical if there exists a bipartite graph G with partite sets V1 and V2 such that s1 and s2 are the degrees in G of the vertices in V1 and V2, respectively. In this paper, we introduce the concept of 1-graphical sequences and present a necessary and sufficient condition for a sequence to be 1-graphical in terms of bigraphical sequences.

Spectral Properties of Unitary Cayley Graphs of Finite Commutative Rings

The Electronic Journal of Combinatorics ◽

10.37236/2390 ◽

2012 ◽

Vol 19 (4) ◽

Cited By ~ 10

Author(s):

Xiaogang Liu ◽

Sanming Zhou

Keyword(s):

Spectral Properties ◽

Line Graph ◽

Commutative Rings ◽

Sufficient Condition ◽

Absolute Value ◽

Vertex Set ◽

Finite Commutative Ring ◽

Unitary Cayley Graph ◽

Let $R$ be a finite commutative ring. The unitary Cayley graph of $R$, denoted $G_R$, is the graph with vertex set $R$ and edge set $\left\{\{a,b\}:a,b\in R, a-b\in R^\times\right\}$, where $R^\times$ is the set of units of $R$. An $r$-regular graph is Ramanujan if the absolute value of every eigenvalue of it other than $\pm r$ is at most $2\sqrt{r-1}$. In this paper we give a necessary and sufficient condition for $G_R$ to be Ramanujan, and a necessary and sufficient condition for the complement of $G_R$ to be Ramanujan. We also determine the energy of the line graph of $G_R$, and compute the spectral moments of $G_R$ and its line graph.

IEICE Transactions on Fundamentals of Electronics Communications and Computer Sciences ◽

A Necessary and Sufficient Condition of Supply and Threshold Voltages in CMOS Circuits for Minimum Energy Point Operation

10.1587/transfun.e100.a.2764 ◽

2017 ◽

Vol E100.A (12) ◽

pp. 2764-2775 ◽

Cited By ~ 1

Author(s):

Jun SHIOMI ◽

Tohru ISHIHARA ◽

Hidetoshi ONODERA

Keyword(s):

Minimum Energy ◽

Cmos Circuits ◽

Sufficient Condition ◽

Energy Point ◽

Threshold Voltages ◽

Minimum Energy Point

A Necessary and Sufficient Condition for a Local Commutative Algebra to be a Moduli Algebra: Weighted Homogeneous Case

Complex Analytic Singularities ◽

10.2969/aspm/00810687 ◽

2018 ◽

Author(s):

Stephen S.-T. Yau

Keyword(s):

Commutative Algebra ◽

Sufficient Condition ◽

Homogeneous Case ◽

Moduli Algebra ◽

A Proposed Framework Emphasizing Auditor Reliability over Auditor Independence

Accounting Horizons ◽

10.2308/acch.2003.17.3.257 ◽

2003 ◽

Vol 17 (3) ◽

pp. 257-266 ◽

Cited By ~ 31

Author(s):

Mark H. Taylor ◽

F. Todd DeZoort ◽

Edward Munn ◽

Martha Wetterhall Thomas

Keyword(s):

Public Interest ◽

Auditor Independence ◽

Public Confidence ◽

Sufficient Condition ◽

Accounting Profession ◽

The Public ◽

This paper introduces an auditor reliability framework that repositions the role of auditor independence in the accounting profession. The framework is motivated in part by widespread confusion about independence and the auditing profession's continuing problems with managing independence and inspiring public confidence. We use philosophical, theoretical, and professional arguments to argue that the public interest will be best served by reprioritizing professional and ethical objectives to establish reliability in fact and appearance as the cornerstone of the profession, rather than relationship-based independence in fact and appearance. This revised framework requires three foundation elements to control subjectivity in auditors' judgments and decisions: independence, integrity, and expertise. Each element is a necessary but not sufficient condition for maximizing objectivity. Objectivity, in turn, is a necessary and sufficient condition for achieving and maintaining reliability in fact and appearance.