Clique number
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Some Extremal Graphs with Respect to Sombor Index Open
Let G be a graph with set of vertices V(G)(|V(G)|=n) and edge set E(G). Very recently, a new degree-based molecular structure descriptor, called Sombor index is denoted by SO(G) and is defined as SO=SO(G)=∑vivj∈E(G)dG(vi)2+dG(vj)2, where d…
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On the character degree graph of solvable groups Open
Let \\(G\\) be a finite solvable group, and let \\(\\Delta(G)\\) denote the\n\\emph{prime graph} built on the set of degrees of the irreducible complex\ncharacters of \\(G\\). A fundamental result by P.P. P\\'alfy asserts that the\ncomplem…
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Circle graphs are quadratically χ‐bounded Open
We prove that the chromatic number of a circle graph with clique number $\omega$ is at most $7\omega^2$.
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Coloring graphs with no induced five‐vertex path or gem Open
For a graph , let and , respectively, denote the chromatic number and clique number of . We give an explicit structural description of (, gem)‐free graphs, and show that every such graph satisfies . Moreover, this bound is best possible. H…
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A survey of $\chi$-boundedness Open
If a graph has bounded clique number and sufficiently large chromatic number, what can we say about its induced subgraphs? András Gyárfás made a number of challenging conjectures about this in the early 1980s, which have remained open unti…
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The Clique Numbers and Chromatic Numbers of The Coprime Graph of a Dihedral Group Open
The graph has many properties and characterizations. One interesting topic to discuss is the clique numbers and chromatic numbers. This research will determine the clique numbers and chromatic numbers of the coprime graph of the dihedral g…
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A counterexample to a conjecture about triangle-free induced subgraphs\n of graphs with large chromatic number Open
We prove that for every $n$, there is a graph $G$ with $\\chi(G) \\geq n$ and\n$\\omega(G) \\leq 3$ such that every induced subgraph $H$ of $G$ with $\\omega(H)\n\\leq 2$ satisfies $\\chi(H) \\leq 4$.\n This disproves a well-known conjectu…
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General multiplicative Zagreb indices of graphs with given clique number Open
We obtain lower and upper bounds on general multiplicative Zagreb indices for graphs of given clique number and order. Bounds on the basic multiplicative Zagreb indices and on the multiplicative sum Zagreb index follow from our results. We…
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Polynomial bounds for chromatic number II: Excluding a star‐forest Open
The Gyárfás–Sumner conjecture says that for every forest , there is a function such that if is ‐free then (where are the chromatic number and the clique number of ). Louis Esperet conjectured that, whenever such a statement holds, can be c…
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Prime ideal sum graph of a commutative ring Open
Let [Formula: see text] be a commutative ring with identity. The prime ideal sum graph of [Formula: see text], denoted by [Formula: see text], is a graph whose vertices are nonzero proper ideals of [Formula: see text] and two distinct vert…
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On the eigenvalues of zero-divisor graph associated to finite commutative ring Open
Let Z(R) be the set of zero-divisors of a commutative ring R with non-zero identity and be the set of non-zero zero-divisors of R. The zero-divisor graph of R, denoted by is a simple graph whose vertex set is and two vertices are adjacent …
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Cliques in rank-1 random graphs: The role of inhomogeneity Open
We study the asymptotic behavior of the clique number in rank-1 inhomogeneous\nrandom graphs, where edge probabilities between vertices are roughly\nproportional to the product of their vertex weights. We show that the clique\nnumber is co…
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Clean graph of a ring Open
Let [Formula: see text] be a ring (not necessarily commutative) with identity. The clean graph [Formula: see text] of a ring [Formula: see text] is a graph with vertices in form [Formula: see text], where [Formula: see text] is an idempote…
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A BOUND FOR THE CHROMATIC NUMBER OF (, GEM)-FREE GRAPHS Open
As usual, $P_{n}$ ( $n\geq 1$ ) denotes the path on $n$ vertices. The gem is the graph consisting of a $P_{4}$ together with an additional vertex adjacent to each vertex of the $P_{4}$ . A graph is called ( $P_{5}$ , gem)-free if it has no…
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The Clique Number and The Chromatics Number Of The Coprime Graph for The Generalized Quarternion Group Open
Graph theory can give a representation of abstract mathematical systems such as groups or rings. We have many graph representations for a group, in this study we use the coprime graph representation for a generalized quaternion group to fi…
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Relations between the energy of graphs and other graph parameters Open
In this paper we give various relations between the energy of graphs and other graph parameters as Randić index, clique number, number of vertices and edges, maximum and minimum degree etc. Moreover, new bounds for the energy of complement…
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Some sharp results on the generalized Tur\'an numbers Open
For graphs $T, H$, let $ex(n,T,H)$ denote the maximum number of copies of $T$\nin an $n$-vertex $H$-free graph. In this paper we prove some sharp results on\nthis generalization of Tur\\'an numbers, where our focus is for the graphs $T,H$\…
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Observations on the Lovász θ-Function, Graph Capacity, Eigenvalues, and Strong Products † Open
This paper provides new observations on the Lovász θ-function of graphs. These include a simple closed-form expression of that function for all strongly regular graphs, together with upper and lower bounds on that function for all regular …
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Co-intersection graph of submodules of a module Open
Let M be a unitary left R-module where R is a ring with identity. The co-intersection graph of proper submodules of M, denoted by Ω(M), is an undirected simple graph whose the vertex set V(Ω) is a set of all non-trivial submodules of M and…
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Order Sum Graph of a Group Open
The concept of the order sum graph associated with a finite group based on the order of the group and order of group elements is introduced. Some of the properties and characteristics such as size, chromatic number, domination number, diam…
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Bounding $\chi$ by a fraction of $\Delta$ for graphs without large cliques Open
The greedy coloring algorithm shows that a graph of maximum degree at most $\Delta$ has chromatic number at most $\Delta + 1$, and this is tight for cliques. Much attention has been devoted to improving this "greedy bound" for graphs witho…
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Clique coloring of binomial random graphs Open
A clique coloring of a graph is a coloring of the vertices so that no maximal clique is monochromatic (ignoring isolated vertices). The smallest number of colors in such a coloring is the clique chromatic number. In this paper, we study th…
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The EKR-Module Property of Pseudo-Paley Graphs of Square Order Open
We prove that a family of pseudo-Paley graphs of square order obtained from unions of cyclotomic classes satisfies the Erdős-Ko-Rado (EKR) module property, in a sense that the characteristic vector of each maximum clique is a linear combin…
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Induced Subgraphs of Induced Subgraphs of Large Chromatic Number Open
We prove that, for every graph F with at least one edge, there is a constant $$c_F$$ such that there are graphs of arbitrarily large chromatic number and the same clique number as F in which every F -free induced subgraph has chromatic…
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On the Non-Zero Divisor Graphs of Some Finite Commutative Rings Open
The study of rings and graphs has been explored extensively by researchers. To gain a more effective understanding on the concepts of the rings and graphs, more researches on graphs of different types of rings are required. This manuscript…
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On the spectral characterization of mixed extensions of P<sub>3</sub> Open
A mixed extension of a graph $G$ is a graph $H$ obtained from $G$ by replacing each vertex of $G$ by a clique or a coclique, whilst two vertices in $H$ corresponding to distinct vertices $x$ and $y$ of $G$ are adjacent whenever $x$ and $y$…
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An Application of Hoffman Graphs for Spectral Characterizations of Graphs Open
In this paper, we present the first application of Hoffman graphs for spectral characterizations of graphs. In particular, we show that the 2-clique extension of the $(t+1)\times (t+1)$-grid is determined by its spectrum when $t$ is large …
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ON THE COMPLEMENT OF THE ZERO-DIVISOR GRAPH OF A PARTIALLY ORDERED SET Open
In this paper, it is proved that the complement of the zero-divisor graph of a partially ordered set is weakly perfect if it has finite clique number, completely answering the question raised by Joshi and Khiste [‘Complement of the zero di…
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Prime ideal graphs of commutative rings Open
Let R be a finite commutative ring with identity and P be a prime ideal of R. The vertex set is R - {0} and two distinct vertices are adjacent if their product in P. This graph is called the prime ideal graph of R and denoted by ΓP. The re…
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Optimal chromatic bound for (P2+P3,P2+P3¯ ${P}_{2}+{P}_{3},\bar{{P}_{2}+{P}_{3}}$)‐free graphs Open
For a graph , let () denote its chromatic (clique) number. A is the graph obtained by taking the disjoint union of a two‐vertex path and a three‐vertex path . A is the complement graph of a . In this paper, we study the class of ()‐free gr…