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Let $G$ be the weighted digraph without loops whose vertex set coincides with the set of states of the Markov chain and arc weights are equal to the corresponding transition probabilities. We give a graph-theoretic interpretation to $m_{ij}$. Namely, We show that $m_{ij}=f_{ij}/q_j$ if $i\\ne j$ and $m_{ij}=1/\\tilde q_j$ if $i=j$, where $f_{ij}$ is the total weight of 2-tree spanning converging forests in $G$ that have one tree containing $i$ and t"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"math/0701359","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2007-01-12T19:36:09Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"d4e5122a0b7af7708ba15b2898ab82bdbfe18fab62ab75afacc5760c1c06eb94","abstract_canon_sha256":"7c3c6a0bc65e7c976213c38c91b32ca7646b041b93abe1bf783d56b0f9541149"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:27:13.890288Z","signature_b64":"J5Fi7nWyUk5PmRzcb72srJSx8VFuUbLglvg8vAku35FlN/xtiCyrrUNAFpuDMkCO4a4VOy9ub0I7jjgMpaDYBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c705f9ad6f4202cacb199074870ba7730b314ce6e3e68c84f96d138d9f16f99f","last_reissued_at":"2026-05-18T00:27:13.889860Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:27:13.889860Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A graph theoretic interpretation of the mean first passage times","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.PR","authors_text":"Pavel Chebotarev","submitted_at":"2007-01-12T19:36:09Z","abstract_excerpt":"Let $m_{ij}$ be the mean first passage time from state $i$ to state $j$ in an $n$-state ergodic homogeneous Markov chain with transition matrix $T$. 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