• Journal of the Korean Mathematical Society
• /
• v.50 no.2
• /
• pp.241-257
• /
• 2013

An automorphism group of a graph is said to be $s$-regular if it acts regularly on the set of $s$-arcs in the graph. A graph is $s$-regular if its full automorphism group is $s$-regular. In the present paper, all $s$-regular cubic graphs of order $10p^3$ are classified for each $s{\geq}1$ and each prime $p$.

• Bulletin of the Korean Mathematical Society
• /
• v.43 no.1
• /
• pp.125-140
• /
• 2006

Let $g(n,\;l_1,\;l_2,\;d,\;t,\;q)$ be the number of general4-regular graphs on n labelled vertices with $l_1+2l_2$ loops, d double edges, t triple edges and q quartet edges. We use inclusion and exclusion with five types of properties to determine the asymptotic behavior of $g(n,\;l_1,\;l_2,\;d,\;t,\;q)$ and hence that of g(2n), the total number of general 4-regular graphs where $l_1,\;l_2,\;d,\;t\;and\;q\;=\;o(\sqrt{n})$, respectively. We show that almost all general 4-regular graphs are 2-connected. Moreover, we determine the asymptotic numbers of general 4-regular graphs with given connectivities.

• Kyungpook Mathematical Journal
• /
• v.64 no.1
• /
• pp.171-184
• /
• 2024

In this paper, we show that if G is strongly regular then the Gallai graph Γ(G) and the anti-Gallai graph ∆(G) of G are edge-regular. We also identify conditions under which the Gallai and anti-Gallai graphs are themselves strongly regular, as well as conditions under which they are 2-connected. We include also a number of concrete examples and a discussion of spectral properties of the Gallai and anti-Gallai graphs.

• Honam Mathematical Journal
• /
• v.27 no.2
• /
• pp.183-194
• /
• 2005

A permutation graph is the graph of inversions in a permutation. Here we determine whether a given labelled graph is a permutation graph or not and when a graph is a permutation graph we find the associated permutation. We also characterize all the 2-regular permutation graphs.

• Bulletin of the Korean Mathematical Society
• /
• v.37 no.3
• /
• pp.487-491
• /
• 2000

Let G be a covering graph of G. We show that the line graph of G covers the line graph of G. Moreover, if the first covering is regular, then the line-graph covering is regular.

• Kyungpook Mathematical Journal
• /
• v.61 no.3
• /
• pp.671-677
• /
• 2021

For a distance-regular graph with valency k, second largest eigenvalue r and diameter D, it is known that r ≥ $min\{\frac{{\lambda}+\sqrt{{\lambda}^2+4k}}{2},\;a_3\}$ if D = 3 and r ≥ $\frac{{\lambda}+\sqrt{{\lambda}^2+4k}}{2}$ if D ≥ 4, where λ = a₁. This result can be generalized to the class of edge-regular graphs. For an edge-regular graph with parameters (v, k, λ) and diameter D ≥ 4, we compare $\frac{{\lambda}+\sqrt{{\lambda}^2+4k}}{2}$ with the local valency λ to find a relationship between the second largest eigenvalue and the local valency. For an edge-regular graph with diameter 3, we look at the number $\frac{{\lambda}-\bar{\mu}+\sqrt{({\lambda}-\bar{\mu})^2+4(k-\bar{\mu})}}{2}$, where $\bar{\mu}=\frac{k(k-1-{\lambda})}{v-k-1}$, and compare this number with the local valency λ to give a relationship between the second largest eigenvalue and the local valency. Also, we apply these relationships to distance-regular graphs.

• Journal of applied mathematics & informatics
• /
• v.31 no.5_6
• /
• pp.651-659
• /
• 2013

A graph is $s$-regular if its automorphism group acts regularly on the set of its $s$-arcs. In this paper, the cubic $s$-regular graphs of order 12p, 36p, 44p, 52p, 66p, 68p and 76p are classified for each $s{\geq}1$ and each prime $p$. The number of cubic $s$-regular graphs of order 12p, 36p, 44p, 52p, 66p, 68p and 76p is 4, 3, 7, 8, 1, 4 and 1, respectively. As a partial result, we determine all cubic $s$-regular graphs of order 70p except for $p$ = 31, 41.

• Journal of applied mathematics & informatics
• /
• v.35 no.1_2
• /
• pp.95-111
• /
• 2017

Recently, interval-valued fuzzy graph is a growing research topic as it is the generalization of fuzzy graphs. The interval-valued fuzzy graphs are more flexible and compatible than fuzzy graphs due to the fact that they allowed the degree of membership of a vertex to an edge to be represented by interval values in [0.1] rather than the crisp values between 0 and 1. In this paper, we introduce the concepts of regular and totally regular interval-valued fuzzy graphs and discusses some properties of the ${\mu}$-complement of interval-valued fuzzy graph. Self ${\mu}$-complementary interval-valued fuzzy graphs and self-weak ${\mu}$-complementary interval-valued fuzzy graphs are defined and a necessary condition for an interval valued fuzzy graph to be self ${\mu}$-complementary is discussed. We define busy vertices and free vertices in interval valued fuzzy graph and study their image under an isomorphism.

• Kyungpook Mathematical Journal
• /
• v.60 no.2
• /
• pp.349-359
• /
• 2020

We study perfect 2-colorings of regular graphs. In particular, we consider the 4-regular case. We obtain a characterization of perfect 2-colorings of toroidal grids.

• Kyungpook Mathematical Journal
• /
• v.48 no.2
• /
• pp.331-335
• /
• 2008

In this paper, we investigate the number of ones in rows of Pascal's Rectangle. Using these results, we determine the existence of regular bipartite Steinhaus graphs. Also, we give an upper bound for the minimum degree of bipartite Steinhaus graphs.