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Given the adjacency matrix $\mathbf{A}$ for a simple connected graph, the characteristic polynomial is defined as: $$ p(\lambda) = \det(\lambda \mathbf{I} - \mathbf{A})$$

Now if an edge between vertex $i$ and $j$ is "labelled" with another variable $x$, then we could consider a bivariate polynomial:

$$ f(\lambda,x) = \det(\lambda \mathbf{I} + x (\mathbf{e}_{ij}+\mathbf{e}_{ji}) - \mathbf{A})$$

where $\mathbf{e}_{ij}$ is the matrix with all entries zero except row $i$, column $j$, which is 1.

Or similarly, if the vertex $i$ was labelled with the variable $x$, we could consider the polynomial:

$$ g(\lambda,x) = \det(\lambda \mathbf{I} + x \mathbf{e}_{ii} - \mathbf{A})$$

This of course could be extended to more labels, but I'm mostly interested in graphs with a single label at the moment.

Have these been studied before, and is there a name for these kinds of polynomials?

I'm particularly interested in learning about simple connected graphs which are non-isomorphic, and have the same "labelled characteristic polynomial", but differ only by a single labelled edge like $$\det(\lambda \mathbf{I} + x(\mathbf{e}_{ij}+\mathbf{e}_{ji}) - A) = \det(\lambda \mathbf{I} + x(\mathbf{e}_{mn}+\mathbf{e}_{nm}) - A)$$ where $mn$ and $ij$ are distinct edges. I'm assuming that is possible given the difficulty of graph isomorphism, but I'm not sure how to go about constructing such graphs.

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    $\begingroup$ I recommend you to ask this question at MathOverflow (MO), because it seems too specific for MSE and also chances to obtain a qualified answer look better at MO. $\endgroup$ Aug 5 '16 at 17:05
  • $\begingroup$ @AlexRavsky I don't really understand the relation between the two sites. I kind of got the impression that MO was for serious researchers. I dont even have a math degree. I'm just curious about math. $\endgroup$ Aug 6 '16 at 0:21
  • $\begingroup$ Yes, you are right, but your question is specific. For instance, I have a math degree and wrote some papers in graph theory, but I don't know an answer to your question. $\endgroup$ Aug 6 '16 at 9:19
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    $\begingroup$ Did you ask this on MO then? I'm interested too! Also, I haven't had time to properly look at it, but a lot on graph labelling in here (and a tremendous compilation of references): combinatorics.org/ojs/index.php/eljc/article/viewFile/DS6/pdf $\endgroup$
    – kbau
    Aug 8 '16 at 10:43
  • $\begingroup$ @kbau I finally asked on MO mathoverflow.net/questions/247416/… I'm still digesting the references Alex gave, but based on his answer I think I should have said "color" instead of "label" a vertex/edge. $\endgroup$ Aug 13 '16 at 7:13
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OK, if you wish to obtain an answer here then I confirm my above comments that your should talk with a specialist. My google search extended my knowledge of an answer not far from zero.

I didn’t find exactly the same generalization of a characteristic polynomial of a graph as you proposed but I guess there should be some of them. See, for instance a paper “A Generalization of the Characteristic Polynomial of a Graph” by Richard J. Lipton and Nisheeth K. Vishnoi.

Personally I see two natural directions in which a further generalization may be useful. The first is a problem of an isomorphism of vertex or edge colored graphs. The second is a problem of a simultaneous isomorphism of two graphs with common vertex set (If I remember it right, a corresponding problem for the similarity of pairs of matrices is so called “wild problem” and when specialists in matrices encounter an equivalent problem they said with respect: “O-o, so it is the wild problem” and stop. Here is even written (in Russian) that it is not expected to obtain a reasonable answer to such a problem).

how to go about constructing non-isomorphic graphs with matching characteristic polynomial with a single "edge label".

I suggest such graphs exist, because the quest to find two non-isomorphic graphs with matching characteristic polynomials (so called cospectral graphs) and even more additional restrictions, looks elaborated. See, for instance, the following papers:

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