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Motivated by Stanley's conjecture about the $e$-positivity of claw-free incomparability graphs, Hamel and her collaborators studied the $e$-positivity of $(claw, H)$-free graphs, where $H$ is a four-vertex graph. In this paper we establish…

In 1995 Stanley conjectured that the chromatic symmetric functions of the graphs $P_{d,2}$, which we call triangular ladders, were $e$-positive. In this paper we confirm this conjecture, which is also an unsolved case of the celebrated…

We study "positive" graphs that have a nonnegative homomorphism number into every edge-weighted graph (where the edgeweights may be negative). We conjecture that all positive graphs can be obtained by taking two copies of an arbitrary…

We prove Gebhard and Sagan's $(e)$-positivity of the line graphs of tadpoles in noncommuting variables. This implies the $e$-positivity of these line graphs. We then extend this $(e)$-positivity result to that of certain cycle-chord graphs,…

We describe a way to decompose the chromatic symmetric function as a positive sum of smaller pieces. We show that these pieces are $e$-positive for cycles. Then we prove that attaching a cycle to a graph preserves the $e$-positivity of…

Motivated by Stanley's $\mathbf{(3+1)}$-free conjecture on chromatic symmetric functions, Foley, Ho\`{a}ng and Merkel introduced the concept of strong $e$-positivity and conjectured that a graph is strongly $e$-positive if and only if it is…

A graph is $n$-e.c. ($n$-existentially closed) if for every pair of subsets $A, B$ of vertex set $V$ of the graph such that $A \cap B = \emptyset$ and $|A| + |B| = n$, there is a vertex $z$ not in $A \cup B$ joined to each vertex of $A$ and…

In the mid-1990s, Stanley and Stembridge conjectured that the chromatic symmetric functions of claw-free co-comparability (also called incomparability) graphs were e-positive. The quest for the proof of this conjecture has led to an…

We introduce two classes of graphs - suns and dumbbells, both with few variations and explore their chromatic symmetric function and its $e$-positivity. We also give many connections of these two classes with other classes of connected…

In this paper, we introduce a family of tetravalent graphs called propeller graphs, denoted by $Pr_{n}(b,c,d)$. We then produce three infinite subfamilies and one finite subfamily of arc-transitive propeller graphs, and show that all such…

We establish the $e$-positivity of cycle-chord graphs by using the composition method which is developed by Zhou and the author recently. Our method is simpler than the $(e)$-positivity approach which is used for handling cycle-chords with…

In this paper, we study positivity phenomena for the $e$-coefficients of Stanley's chromatic function of a graph. We introduce a new combinatorial object: the {\em correct} sequences of unit interval orders, and using these, in certain…

In this work, we define an orthogonal graph on the set of equivalence classes of $(2\nu + \delta)-$tuples over $\mathbb{Z}_{2^n}$ where $n$ and $\nu$ are positive integers and $\delta = 0, 1$ or $2$. We classify our graph if it is strongly…

Let ${\cal G}=(G,w) $ be a positive-weighted graph, that is a graph $G$ endowed with a function $w$ from the edge set of $G$ to the set of positive real numbers; for any distinct vertices $i,j $, we define $D_{i,j}({\cal G})$ to be the…

In a study, published in Nature, researchers from DeepMind and mathematicians demonstrated a general framework using machine learning to make conjectures in pure mathematics. Here, we build upon this framework to develop a method for…

A graph G=(V,E) is a pairwise compatibility graph (PCG) if there exists an edge-weighted tree T and two non-negative real numbers `d' and `D' such that each leaf `u' of T is a node of V and the edge `(u,v) belongs to E' iff `d <= d_T(u, v)…

Let $s$ be a positive integer. A graph is $s$-transitive if its automorphism group is transitive on s-arcs but not on $(s + 1)$-arcs. In this paper, we study all tetravalent s-transitive graphs of order $6p^2$.

A graph $H$ is said to be positive if the homomorphism density $t_H(G)$ is non-negative for all weighted graphs $G$. The positive graph conjecture proposes a characterisation of such graphs, saying that a graph is positive if and only if it…

A class of graphs is bridge-addable if given a graph $G$ in the class, any graph obtained by adding an edge between two connected components of $G$ is also in the class. The authors recently proved a conjecture of McDiarmid, Steger, and…

A $\mathbb{T}$-gain graph is a triple $\Phi=(G,\mathbb{T},\varphi)$ consisting of a graph $G=(V,E)$, the circle group $\mathbb{T}=\{z\in C: |z|=1\}$ and a gain function $\varphi:\overrightarrow{E}\rightarrow \mathbb{T}$ such that…