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Dears Let $G$ be a graph with $n$ vertices and let $T(G)$ be the set of all bijections from vertices $V(G)$ of $G$ to the set $\{1,\dots,n\}$. Let $E(G)$ be as ususal the set of edges of $G$. Is the …

Let $G$ be a graph and $e$ be an edge of the graph $G$ such that the subgraph $G\setminus e$ is connected. The subgraph $G\setminus e$ is the subgraph of $G$ obtained by deletion of the edge $e$ of $G …

This is answered in the following recent paper: The graphs with all but two eigenvalues equal to ±1 By Sebastian M. Cioabă, Willem H. Haemers, Jason Vermette, Wiseley Wong One may download it from …

Let $G$ be a graph without any hole or antihole of odd length at least 5 (i.e. $G$ is a Berge graph and so by the Strong Perfect Graph Theorem, $G$ is perfect). Assume further that $G$ has no antihol …

I have proposed very recently a question in the following link concerning the title of the current question: Difference of the maximum eigenvalue of a graph with the one of one-edge-deleted subgraph …

It is well-known that the Paley graph $P(q)$ is a strongly regular graph with parameters $(4t+1,2t,t-1,t)$. Suppose that $v$ is a vertex in the Paley graph $P(q)$. Suppoe that $C$ is the set of all …

Let $n>15$ and $A=A_n$ be the alternating group of degree $n$ on the set $\{1,\dots,n\}$. For any subset $X$ of $\{1,\dots,n\}$, $Stab_A(X)$ denote as usual the set of all permutations $\sigma$ of $A …

Let $G$ be a regular graph having spanning regular subgraphs $G_1,\dots, G_k$ whose edge sets are disjoint and their union is the whole edge set of $G$. Is it true that the clique number of $G$ is bou …

Let $n>15$ be an integer. Suppose also $n=\sum_{i=1}^n ic_i$, where $c_i$ are non-negative integers. Assume further that $c_1<4$. Is the following inequality true? $$\frac{n!}{\prod_{i=1}^{n}i^{c_i}\ …

Let $n\geq 1$ be an integer. The Friendship Graph (or Dutch windmill graph or $n$-Fan) $F_n$ is a graph that can be constructed by coalescence $n$ copies of the cycle graph $C_3$ with a common vertex. …

It is known that the Paley graph $P(q)$ for $q = 5, 9, 13$ or $17$ vertices are the only strongly regular graph with the parameters as $P(q)$. If $q \geq 25$, is the following assertion true: The …

Let $n>3$ be an odd integer and let $K_n$ denote the complete graph on $n$ vertices. For which integers $n$ the line graph $L(K_n)$ is a Cayley graph? For even $n$, it follows from a result of Watkins …

Is there a recursive formulae to compute the edge chromatic polynomial of a graph? The following formulae is known for the vertex chromatic polynomial of a grapg $G$ $P(G,x)=P(G-uv, x)- P(G/uv,x)$ …

Let $p$ be a prime and $\mathbb{Z}_p$ denote as usual the field of order $p$. Let $AL(p)$ be the affine linear group $\{x\mapsto ax+b \;|\; a\in \mathbb{Z}_p\setminus \{0\}, b\in\mathbb{Z}_p\}$. For a …

You may start with the PhD thesis of P. P. Campbell http://www-circa.mcs.st-and.ac.uk/Theses/ppcphd.pdf