scholarly journals Signed graphs connected with the root lattice

BIBECHANA ◽  
2014 ◽  
Vol 11 ◽  
pp. 157-160
Author(s):  
RN Yadav
Keyword(s):  
Connected Graph ◽  
Signed Graph ◽  
Connected Components ◽  
Root Lattice ◽  
Signed Graphs ◽  
Cut Vertex ◽  
Vertex Set

For any base of the root lattice (An) we can construct a signed graph. A signed graph is one whose edges are signed by +1 or -1. A signed graph is balanced if and only if its vertex set can be divided into two sets-either of which may be empty–so that each edge between the sets is negative and each edge within a set is positive. For a given signed graph Tsaranov, Siedel and Cameron constructed the corresponding root lattice. In the present work we have dealt with signed graphs corresponding to the root lattice An. A connected graph is called a Fushimi tree if its all blocks are complete subgraphs. A Fushimi tree is said to be simple when by deleting any cut vertex we have always two connected components. A signed Fushimi tree is called a Fushimi tree with standard sign if it can be transformed into a signed Fushimi tree whose all edges are signed by +1 by switching. Here we have proved that any signed graph corresponding to An is a simple Fushimi tree with standard sign. Our main result is that s simple Fushimi tree with standard sign is contained in the cluster given by a line. DOI: http://dx.doi.org/10.3126/bibechana.v11i0.10396 BIBECHANA 11(1) (2014) 157-160

On The Unitary Cayley Ring Signed Graphs$S_n^\oplus$

2013 ◽  
Vol 14 (04) ◽  
pp. 1350020 ◽  
Author(s):  
DEEPA SINHA ◽  
AYUSHI DHAMA
Keyword(s):  
Positive Integer ◽  
Cayley Graph ◽  
Signed Graph ◽  
Signed Graphs ◽  
Vertex Set ◽  
Unitary Cayley Graph

A Signed graph (or sigraph in short) is an ordered pair S = (G, σ), where G is a graph G = (V, E) and σ : E → {+, −} is a function from the edge set E of G into the set {+, −}. For a positive integer n > 1, the unitary Cayley graph Xnis the graph whose vertex set is Zn, the integers modulo n and if Undenotes set of all units of the ring Zn, then two vertices a, b are adjacent if and only if a − b ∈ Un. In this paper, we have obtained a characterization of balanced and clusterable unitary Cayley ring sigraph [Formula: see text]. Further, we have established a characterization of canonically consistent unitary Cayley ring sigraph [Formula: see text], where n has at most two distinct odd primes factors. Also sign-compatibility has been worked out for the same.

scholarly journals On the Unitary Cayley Signed Graphs

10.37236/716 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Deepa Sinha ◽  
Pravin Garg
Keyword(s):  
Positive Integer ◽  
Cayley Graph ◽  
Signed Graph ◽  
Signed Graphs ◽  
Prime Factors ◽  
Vertex Set ◽  
Unitary Cayley Graph ◽  
Sigma E

A $signed graph$ (or $sigraph$ in short) is an ordered pair $S = (S^u, \sigma)$, where $S^u$ is a graph $G = (V, E)$ and $\sigma : E\rightarrow \{+,-\}$ is a function from the edge set $E$ of $S^u$ into the set $\{+, -\}$. For a positive integer $n > 1$, the unitary Cayley graph $X_n$ is the graph whose vertex set is $Z_n$, the integers modulo $n$ and if $U_n$ denotes set of all units of the ring $Z_n$, then two vertices $a, b$ are adjacent if and only if $a-b \in U_n$. For a positive integer $n > 1$, the unitary Cayley sigraph $\mathcal{S}_n = (\mathcal{S}^u_n, \sigma)$ is defined as the sigraph, where $\mathcal{S}^u_n$ is the unitary Cayley graph and for an edge $ab$ of $\mathcal{S}_n$, $$\sigma(ab) = \begin{cases} + & \text{if } a \in U_n \text{ or } b \in U_n,\\ - & \text{otherwise.} \end{cases}$$ In this paper, we have obtained a characterization of balanced unitary Cayley sigraphs. Further, we have established a characterization of canonically consistent unitary Cayley sigraphs $\mathcal{S}_n$, where $n$ has at most two distinct odd prime factors.

scholarly journals Characterization of 2-Path Product Signed Graphs with Its Properties

2017 ◽  
Vol 2017 ◽  
pp. 1-8 ◽  
Author(s):  
Deepa Sinha ◽  
Deepakshi Sharma
Keyword(s):  
Social Sciences ◽  
Simple Graph ◽  
Signed Graph ◽  
Signed Graphs ◽  
Positive Edge ◽  
Friendly Relation ◽  
Negative Edge ◽  
Vertex Set

A signed graph is a simple graph where each edge receives a sign positive or negative. Such graphs are mainly used in social sciences where individuals represent vertices friendly relation between them as a positive edge and enmity as a negative edge. In signed graphs, we define these relationships (edges) as of friendship (“+” edge) or hostility (“-” edge). A 2-path product signed graph S#^S of a signed graph S is defined as follows: the vertex set is the same as S and two vertices are adjacent if and only if there exists a path of length two between them in S. The sign of an edge is the product of marks of vertices in S where the mark of vertex u in S is the product of signs of all edges incident to the vertex. In this paper, we give a characterization of 2-path product signed graphs. Also, some other properties such as sign-compatibility and canonically-sign-compatibility of 2-path product signed graphs are discussed along with isomorphism and switching equivalence of this signed graph with 2-path signed graph.

scholarly journals A New Bound on the Domination Number of Graphs with Minimum Degree Two

10.37236/499 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Michael A. Henning ◽  
Ingo Schiermeyer ◽  
Anders Yeo
Keyword(s):  
Connected Graph ◽  
Minimum Degree ◽  
Domination Number ◽  
Degree Condition ◽  
Cut Vertex ◽  
Delta G ◽  
Vertex Disjoint ◽  
Special Cycle

For a graph $G$, let $\gamma(G)$ denote the domination number of $G$ and let $\delta(G)$ denote the minimum degree among the vertices of $G$. A vertex $x$ is called a bad-cut-vertex of $G$ if $G-x$ contains a component, $C_x$, which is an induced $4$-cycle and $x$ is adjacent to at least one but at most three vertices on $C_x$. A cycle $C$ is called a special-cycle if $C$ is a $5$-cycle in $G$ such that if $u$ and $v$ are consecutive vertices on $C$, then at least one of $u$ and $v$ has degree $2$ in $G$. We let ${\rm bc}(G)$ denote the number of bad-cut-vertices in $G$, and ${\rm sc}(G)$ the maximum number of vertex disjoint special-cycles in $G$ that contain no bad-cut-vertices. We say that a graph is $(C_4,C_5)$-free if it has no induced $4$-cycle or $5$-cycle. Bruce Reed [Paths, stars and the number three. Combin. Probab. Comput. 5 (1996), 277–295] showed that if $G$ is a graph of order $n$ with $\delta(G) \ge 3$, then $\gamma(G) \le 3n/8$. In this paper, we relax the minimum degree condition from three to two. Let $G$ be a connected graph of order $n \ge 14$ with $\delta(G) \ge 2$. As an application of Reed's result, we show that $\gamma(G) \le \frac{1}{8} ( 3n + {\rm sc}(G) + {\rm bc}(G))$. As a consequence of this result, we have that (i) $\gamma(G) \le 2n/5$; (ii) if $G$ contains no special-cycle and no bad-cut-vertex, then $\gamma(G) \le 3n/8$; (iii) if $G$ is $(C_4,C_5)$-free, then $\gamma(G) \le 3n/8$; (iv) if $G$ is $2$-connected and $d_G(u) + d_G(v) \ge 5$ for every two adjacent vertices $u$ and $v$, then $\gamma(G) \le 3n/8$. All bounds are sharp.

The Maximum Genus of Cartesian Products of Graphs

1974 ◽  
Vol 26 (5) ◽  
pp. 1025-1035 ◽  
Author(s):  
Joseph Zaks
Keyword(s):  
Connected Graph ◽  
Connected Components ◽  
Open Disk ◽  
Maximum Genus ◽  
Cartesian Products ◽  
Euler’S Formula ◽  
Cartesian Products Of Graphs

The maximum genus γM(G) of a connected graph G has been defined in [2] as the maximum g for which there exists an embedding h : G —> S(g), where S(g) is a compact orientable 2-manifold of genus g, such that each one of the connected components of S(g) — h(G) is homeomorphic to an open disk; such an embedding is called cellular. If G is cellularly embedded in S(g), having V vertices, E edges and F faces, then by Euler's formulaV-E + F = 2-2g.

scholarly journals Metric basis and metric dimension of 1-pentagonal carbon nanocone networks

2020 ◽  
Vol 10 (1) ◽  
Author(s):  
Zafar Hussain ◽  
Mobeen Munir ◽  
Ashfaq Ahmad ◽  
Maqbool Chaudhary ◽  
Junaid Alam Khan ◽  
...  
Keyword(s):  
Combinatorial Chemistry ◽  
Connected Graph ◽  
Cardinal Number ◽  
Metric Dimension ◽  
Molecular Topology ◽  
Resolving Set ◽  
Computer Chemistry ◽  
Carbon Nanocones ◽  
Vertex Set ◽  
Metric Basis

AbstractResolving set and metric basis has become an integral part in combinatorial chemistry and molecular topology. It has a lot of applications in computer, chemistry, pharmacy and mathematical disciplines. A subset S of the vertex set V of a connected graph G resolves G if all vertices of G have different representations with respect to S. A metric basis for G is a resolving set having minimum cardinal number and this cardinal number is called the metric dimension of G. In present work, we find a metric basis and also metric dimension of 1-pentagonal carbon nanocones. We conclude that only three vertices are minimal requirement for the unique identification of all vertices in this network.

A Note on Primitive Graphs

1972 ◽  
Vol 15 (3) ◽  
pp. 437-440 ◽  
Author(s):  
I. Z. Bouwer ◽  
G. F. LeBlanc
Keyword(s):  
Connected Graph ◽  
Common Vertex ◽  
Connected Subgraph ◽  
Property A ◽  
Vertex Set

Let G denote a connected graph with vertex set V(G) and edge set E(G). A subset C of E(G) is called a cutset of G if the graph with vertex set V(G) and edge set E(G)—C is not connected, and C is minimal with respect to this property. A cutset C of G is simple if no two edges of C have a common vertex. The graph G is called primitive if G has no simple cutset but every proper connected subgraph of G with at least one edge has a simple cutset. For any edge e of G, let G—e denote the graph with vertex set V(G) and with edge set E(G)—e.

New bounds on the independence number of connected graphs

2018 ◽  
Vol 10 (05) ◽  
pp. 1850069
Author(s):  
Nader Jafari Rad ◽  
Elahe Sharifi
Keyword(s):  
Lower Bound ◽  
Connected Graph ◽  
Maximum Degree ◽  
Applied Mathematics ◽  
Independence Number ◽  
Independent Set ◽  
Maximum Cardinality ◽  
Connected Graphs ◽  
Cut Vertex

The independence number of a graph [Formula: see text], denoted by [Formula: see text], is the maximum cardinality of an independent set of vertices in [Formula: see text]. [Henning and Löwenstein An improved lower bound on the independence number of a graph, Discrete Applied Mathematics  179 (2014) 120–128.] proved that if a connected graph [Formula: see text] of order [Formula: see text] and size [Formula: see text] does not belong to a specific family of graphs, then [Formula: see text]. In this paper, we strengthen the above bound for connected graphs with maximum degree at least three that have a non-cut-vertex of maximum degree. We show that if a connected graph [Formula: see text] of order [Formula: see text] and size [Formula: see text] has a non-cut-vertex of maximum degree then [Formula: see text], where [Formula: see text] is the maximum degree of the vertices of [Formula: see text]. We also characterize all connected graphs [Formula: see text] of order [Formula: see text] and size [Formula: see text] that have a non-cut-vertex of maximum degree and [Formula: see text].

Graham’s pebbling conjecture holds for the product of a graph and a sufficiently large complete bipartite graph

2019 ◽  
Vol 11 (06) ◽  
pp. 1950068
Author(s):  
Nopparat Pleanmani
Keyword(s):  
Bipartite Graph ◽  
Network Optimization ◽  
Connected Graph ◽  
Complete Bipartite Graph ◽  
Fixed Number ◽  
Arbitrary Vertex ◽  
Graph Pebbling ◽  
Vertex Set ◽  
Graham’S Conjecture ◽  
Pebbling Number

A graph pebbling is a network optimization model for the transmission of consumable resources. A pebbling move on a connected graph [Formula: see text] is the process of removing two pebbles from a vertex and placing one of them on an adjacent vertex after configuration of a fixed number of pebbles on the vertex set of [Formula: see text]. The pebbling number of [Formula: see text], denoted by [Formula: see text], is defined to be the least number of pebbles to guarantee that for any configuration of pebbles on [Formula: see text] and arbitrary vertex [Formula: see text], there is a sequence of pebbling movement that places at least one pebble on [Formula: see text]. For connected graphs [Formula: see text] and [Formula: see text], Graham’s conjecture asserted that [Formula: see text]. In this paper, we show that such conjecture holds when [Formula: see text] is a complete bipartite graph with sufficiently large order in terms of [Formula: see text] and the order of [Formula: see text].

A new graph radio k-coloring algorithm

2019 ◽  
Vol 11 (01) ◽  
pp. 1950005 ◽  
Author(s):  
Laxman Saha ◽  
Pratima Panigrahi
Keyword(s):  
Connected Graph ◽  
Channel Assignment ◽  
Time Complexity ◽  
Radio Spectrum ◽  
Circulant Graph ◽  
Communication Services ◽  
Channel Assignment Problem ◽  
Vertex Set ◽  
General Graphs ◽  
Simple Connected Graph

Due to the rapid growth in the use of wireless communication services and the corresponding scarcity and the high cost of radio spectrum bandwidth, Channel assignment problem (CAP) is becoming highly important. Radio [Formula: see text]-coloring of graphs is a variation of CAP. For a positive integer [Formula: see text], a radio [Formula: see text]-coloring of a simple connected graph [Formula: see text] is a mapping [Formula: see text] from the vertex set [Formula: see text] to the set [Formula: see text] of non-negative integers such that [Formula: see text] for each pair of distinct vertices [Formula: see text] and [Formula: see text] of [Formula: see text], where [Formula: see text] is the distance between [Formula: see text] and [Formula: see text] in [Formula: see text]. The span of a radio [Formula: see text]-coloring [Formula: see text], denoted by [Formula: see text], is defined as [Formula: see text] and the radio[Formula: see text]-chromatic number of [Formula: see text], denoted by [Formula: see text], is [Formula: see text] where the minimum is taken over all radio [Formula: see text]-coloring of [Formula: see text]. In this paper, we present two radio [Formula: see text]-coloring algorithms for general graphs which will produce radio [Formula: see text]-colorings with their spans. For an [Formula: see text]-vertex simple connected graph the time complexity of the both algorithm is of [Formula: see text]. Implementing these algorithms we get the exact value of [Formula: see text] for several graphs (for example, [Formula: see text], [Formula: see text], [Formula: see text], some circulant graph etc.) and many values of [Formula: see text], especially for [Formula: see text].

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