Induced cycle path number of derived graphs

An induced cycle path partition of a simple connected graph [Formula: see text] is a partition of [Formula: see text] into subsets such that the subsets induce cycles and paths. The minimum order taken over all induced cycle path partitions is called the induced cycle path number of [Formula: see text] and is denoted by [Formula: see text]. In this paper, we investigate the induced cycle path number of some derived graphs such as line graph, block graph, subdivision graph, power graph and total graph.

Download Full-text

Hyperbolicity on Graph Operators

Symmetry ◽

10.3390/sym10090360 ◽

2018 ◽

Vol 10 (9) ◽

pp. 360 ◽

Cited By ~ 2

Author(s):

J. Méndez-Bermúdez ◽

Rosalío Reyes ◽

José Rodríguez ◽

José Sigarreta

Keyword(s):

Line Graph ◽

Discrete Mathematics ◽

Total Graph ◽

Subdivision Graph ◽

Gromov Product

A graph operator is a mapping F : Γ → Γ ′ , where Γ and Γ ′ are families of graphs. The different kinds of graph operators are an important topic in Discrete Mathematics and its applications. The symmetry of this operations allows us to prove inequalities relating the hyperbolicity constants of a graph G and its graph operators: line graph, Λ ( G ) ; subdivision graph, S ( G ) ; total graph, T ( G ) ; and the operators R ( G ) and Q ( G ) . In particular, we get relationships such as δ ( G ) ≤ δ ( R ( G ) ) ≤ δ ( G ) + 1 / 2 , δ ( Λ ( G ) ) ≤ δ ( Q ( G ) ) ≤ δ ( Λ ( G ) ) + 1 / 2 , δ ( S ( G ) ) ≤ 2 δ ( R ( G ) ) ≤ δ ( S ( G ) ) + 1 and δ ( R ( G ) ) − 1 / 2 ≤ δ ( Λ ( G ) ) ≤ 5 δ ( R ( G ) ) + 5 / 2 for every graph which is not a tree. Moreover, we also derive some inequalities for the Gromov product and the Gromov product restricted to vertices.

Download Full-text

Reformulated Zagreb Indices of Some Derived Graphs

Mathematics ◽

10.3390/math7040366 ◽

2019 ◽

Vol 7 (4) ◽

pp. 366 ◽

Cited By ~ 6

Author(s):

Jia-Bao Liu ◽

Bahadur Ali ◽

Muhammad Aslam Malik ◽

Hafiz Muhammad Afzal Siddiqui ◽

Muhammad Imran

Keyword(s):

Physical Properties ◽

Chemical Structure ◽

Topological Index ◽

Chemical Reactivity ◽

Line Graph ◽

Total Graph ◽

Zagreb Indices ◽

Subdivision Graph ◽

Chemical Constitution ◽

Vertex Degrees

A topological index is a numeric quantity that is closely related to the chemical constitution to establish the correlation of its chemical structure with chemical reactivity or physical properties. Miličević reformulated the original Zagreb indices in 2004, replacing vertex degrees by edge degrees. In this paper, we established the expressions for the reformulated Zagreb indices of some derived graphs such as a complement, line graph, subdivision graph, edge-semitotal graph, vertex-semitotal graph, total graph, and paraline graph of a graph.

Download Full-text

Distance Degree Index of Some Derived Graphs

Mathematics ◽

10.3390/math7030283 ◽

2019 ◽

Vol 7 (3) ◽

pp. 283 ◽

Cited By ~ 2

Author(s):

Jianzhong Xu ◽

Jia-Bao Liu ◽

Ahsan Bilal ◽

Uzma Ahmad ◽

Hafiz Muhammad Afzal Siddiqui ◽

...

Keyword(s):

Organic Molecules ◽

Line Graph ◽

Chemical Compounds ◽

Topological Indices ◽

Pharmacological Properties ◽

Graph Structure ◽

Total Graph ◽

Graph Vertex ◽

Physical Chemical ◽

Subdivision Graph

Topological indices are numerical values associated with a graph (structure) that can predict many physical, chemical, and pharmacological properties of organic molecules and chemical compounds. The distance degree (DD) index was introduced by Dobrynin and Kochetova in 1994 for characterizing alkanes by an integer. In this paper, we have determined expressions for a DD index of some derived graphs in terms of the parameters of the parent graph. Specifically, we establish expressions for the DD index of a line graph, subdivision graph, vertex-semitotal graph, edge-semitotal graph, total graph, and paraline graph.

Download Full-text

ON THE NORMALISED LAPLACIAN SPECTRUM, DEGREE-KIRCHHOFF INDEX AND SPANNING TREES OF GRAPHS

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972715000027 ◽

2015 ◽

Vol 91 (3) ◽

pp. 353-367 ◽

Cited By ~ 38

Author(s):

JING HUANG ◽

SHUCHAO LI

Keyword(s):

Lower Bound ◽

Characteristic Polynomial ◽

Regular Graph ◽

Spanning Trees ◽

Line Graph ◽

Laplacian Spectrum ◽

Kirchhoff Index ◽

Subdivision Graph ◽

Number Of Spanning Trees ◽

Sharp Lower Bound

Given a connected regular graph $G$, let $l(G)$ be its line graph, $s(G)$ its subdivision graph, $r(G)$ the graph obtained from $G$ by adding a new vertex corresponding to each edge of $G$ and joining each new vertex to the end vertices of the corresponding edge and $q(G)$ the graph obtained from $G$ by inserting a new vertex into every edge of $G$ and new edges joining the pairs of new vertices which lie on adjacent edges of $G$. A formula for the normalised Laplacian characteristic polynomial of $l(G)$ (respectively $s(G),r(G)$ and $q(G)$) in terms of the normalised Laplacian characteristic polynomial of $G$ and the number of vertices and edges of $G$ is developed and used to give a sharp lower bound for the degree-Kirchhoff index and a formula for the number of spanning trees of $l(G)$ (respectively $s(G),r(G)$ and $q(G)$).

Download Full-text

ON DYNAMIC COLORING OF WEB GRAPH

Kongunadu Research Journal ◽

10.26524/krj263 ◽

2018 ◽

Vol 5 (2) ◽

pp. 11-15

Author(s):

Aaresh R.R ◽

Venkatachalam M ◽

Deepa T

Keyword(s):

Chromatic Number ◽

Line Graph ◽

Proper Coloring ◽

Total Graph ◽

Web Graph ◽

Middle Graph

Dynamic coloring of a graph G is a proper coloring. The chromatic number of a graph G is the minimum k such that G has a dynamic coloring with k colors. In this paper we investigate the dynamic chromatic number for the Central graph, Middle graph, Total graph and Line graph of Web graph Wn denoted by C(Wn), M(Wn), T(Wn) and L(Wn) respectively.

Download Full-text

On r-dynamic coloring of comb graphs

Notes on Number Theory and Discrete Mathematics ◽

10.7546/nntdm.2021.27.2.191-200 ◽

2021 ◽

Vol 27 (2) ◽

pp. 191-200

Author(s):

K. Kalaiselvi ◽

◽

N. Mohanapriya ◽

J. Vernold Vivin ◽

◽

...

Keyword(s):

Lower Bound ◽

Chromatic Number ◽

Upper Bound ◽

Line Graph ◽

Proper Coloring ◽

Total Graph ◽

Middle Graph ◽

Different Color

An r-dynamic coloring of a graph G is a proper coloring of G such that every vertex in V(G) has neighbors in at least $\min\{d(v),r\}$ different color classes. The r-dynamic chromatic number of graph G denoted as $\chi_r (G)$, is the least k such that G has a coloring. In this paper we obtain the r-dynamic chromatic number of the central graph, middle graph, total graph, line graph, para-line graph and sub-division graph of the comb graph $P_n\odot K_1$ denoted by $C(P_n\odot K_1), M(P_n\odot K_1), T(P_n\odot K_1), L(P_n\odot K_1), P(P_n\odot K_1)$ and $S(P_n\odot K_1)$ respectively by finding the upper bound and lower bound for the r-dynamic chromatic number of the Comb graph.

Download Full-text

Omega Index of Line and Total Graphs

Journal of Mathematics ◽

10.1155/2021/5552202 ◽

2021 ◽

Vol 2021 ◽

pp. 1-6

Author(s):

Musa Demirci ◽

Sadik Delen ◽

Ahmet Sinan Cevik ◽

Ismail Naci Cangul

Keyword(s):

Line Graph ◽

Graph Invariant ◽

Original Graph ◽

Total Graph ◽

Graph Classes ◽

Number Of Faces

A derived graph is a graph obtained from a given graph according to some predetermined rules. Two of the most frequently used derived graphs are the line graph and the total graph. Calculating some properties of a derived graph helps to calculate the same properties of the original graph. For this reason, the relations between a graph and its derived graphs are always welcomed. A recently introduced graph index which also acts as a graph invariant called omega is used to obtain such relations for line and total graphs. As an illustrative exercise, omega values and the number of faces of the line and total graphs of some frequently used graph classes are calculated.

Download Full-text

Line and Subdivision Graphs Determined by T4-Gain Graphs

Mathematics ◽

10.3390/math7100926 ◽

2019 ◽

Vol 7 (10) ◽

pp. 926 ◽

Cited By ~ 3

Author(s):

Abdullah Alazemi ◽

Milica Anđelić ◽

Francesco Belardo ◽

Maurizio Brunetti ◽

Carlos M. da Fonseca

Keyword(s):

Spectral Properties ◽

Incidence Matrix ◽

Multiplicative Group ◽

Line Graph ◽

Characteristic Polynomials ◽

Signed Graphs ◽

Subdivision Graph ◽

Gain Graphs ◽

The Relationship ◽

Gain Graph

Let T 4 = { ± 1 , ± i } be the subgroup of fourth roots of unity inside T , the multiplicative group of complex units. For a T 4 -gain graph Φ = ( Γ , T 4 , φ ) , we introduce gain functions on its line graph L ( Γ ) and on its subdivision graph S ( Γ ) . The corresponding gain graphs L ( Φ ) and S ( Φ ) are defined up to switching equivalence and generalize the analogous constructions for signed graphs. We discuss some spectral properties of these graphs and in particular we establish the relationship between the Laplacian characteristic polynomial of a gain graph Φ , and the adjacency characteristic polynomials of L ( Φ ) and S ( Φ ) . A suitably defined incidence matrix for T 4 -gain graphs plays an important role in this context.

Download Full-text